Journal of Applied Probability Articles (Project Euclid)
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The latest articles from Journal of Applied Probability on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTTue, 15 Mar 2011 10:10 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Efficient importance sampling in ruin problems for multidimensional regularly varying random walks
http://projecteuclid.org/euclid.jap/1276784893
<strong>Jose Blanchet</strong>, <strong>Jingchen Liu</strong><p><strong>Source: </strong>J. Appl. Probab., Volume 47, Number 2, 301--322.</p><p><strong>Abstract:</strong><br/>
We consider the problem of efficient estimation via simulation of first passage
time probabilities for a multidimensional random walk with heavy-tailed
increments. In addition to being a natural generalization to the problem of
computing ruin probabilities in insurance - in which the focus is the maximum
of a one-dimensional random walk with negative drift - this problem captures
important features of large deviations for multidimensional heavy-tailed
processes (such as the role played by the mean of the process in connection to
the location of the target set). We develop a state-dependent importance
sampling estimator for this class of multidimensional problems. Then, using
techniques based on Lyapunov inequalities, we argue that our estimator is
strongly efficient in the sense that the relative mean squared error of
our estimator can be made arbitrarily small by increasing the number of
replications, uniformly as the probability of interest approaches 0.
</p>projecteuclid.org/euclid.jap/1276784893_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTExtension of de Bruijn's identity to dependent non-Gaussian noise channelshttp://projecteuclid.org/euclid.jap/1466172859<strong>Nayereh Bagheri Khoolenjani</strong>, <strong>Mohammad Hossein Alamatsaz</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 360--368.</p><p><strong>Abstract:</strong><br/>
De Bruijn's identity relates two important concepts in information theory:
Fisher information and differential entropy. Unlike the common practice in the
literature, in this paper we consider general additive non-Gaussian noise
channels where more realistically, the input signal and additive noise are not
independently distributed. It is shown that, for general dependent signal and
noise, the first derivative of the differential entropy is directly related to
the conditional mean estimate of the input. Then, by using Gaussian and
Farlie--Gumbel--Morgenstern copulas, special versions of the result are given
in the respective case of additive normally distributed noise. The previous
result on independent Gaussian noise channels is included as a special case.
Illustrative examples are also provided.
</p>projecteuclid.org/euclid.jap/1466172859_20160617101421Fri, 17 Jun 2016 10:14 EDTExtreme paths in oriented two-dimensional percolationhttp://projecteuclid.org/euclid.jap/1466172860<strong>E. D. Andjel</strong>, <strong>L. F. Gray</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 369--380.</p><p><strong>Abstract:</strong><br/>
A useful result about leftmost and rightmost paths in two-dimensional bond
percolation is proved. This result was introduced without proof in Gray (1991)
in the context of the contact process in continuous time. As discussed here, it
also holds for several related models, including the discrete-time contact
process and two-dimensional site percolation. Among the consequences are a
natural monotonicity in the probability of percolation between different sites
and a somewhat counter-intuitive correlation inequality.
</p>projecteuclid.org/euclid.jap/1466172860_20160617101421Fri, 17 Jun 2016 10:14 EDTAsymptotic behaviour near extinction of continuous-state branching processeshttp://projecteuclid.org/euclid.jap/1466172861<strong>Gabriel Berzunza</strong>, <strong>Juan Carlos Pardo</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 381--391.</p><p><strong>Abstract:</strong><br/>
In this paper we study the asymptotic behaviour near extinction of
(sub-)critical continuous-state branching processes. In particular, we
establish an analogue of Khintchine's law of the iterated logarithm near
extinction time for a continuous-state branching process whose branching
mechanism satisfies a given condition.
</p>projecteuclid.org/euclid.jap/1466172861_20160617101421Fri, 17 Jun 2016 10:14 EDTThe Markov consistency of Archimedean survival processeshttp://projecteuclid.org/euclid.jap/1466172862<strong>J. Jakubowski</strong>, <strong>A. Pytel</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 392--409.</p><p><strong>Abstract:</strong><br/>
In this paper we connect Archimedean survival processes (ASPs) with the theory
of Markov copulas. ASPs were introduced by Hoyle and Meng\"{u}t\"{u}rk (2013)
to model the realized variance of two assets. We present some new properties of
ASPs related to their dependency structure. We study weak and strong Markovian
consistency properties of ASPs. An ASP is weak Markovian consistent, but
generally not strong Markovian consistent. Our results contain necessary and
sufficient conditions for an ASP to be strong Markovian consistent. These
properties are closely related to the concept of Markov copulas, which is very
useful in modelling different dependence phenomena. At the end we present
possible applications.
</p>projecteuclid.org/euclid.jap/1466172862_20160617101421Fri, 17 Jun 2016 10:14 EDTComplexity bounds for Markov chain Monte Carlo algorithms via diffusion limitshttp://projecteuclid.org/euclid.jap/1466172863<strong>Gareth O. Roberts</strong>, <strong>Jeffrey S. Rosenthal</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 410--420.</p><p><strong>Abstract:</strong><br/>
We connect known results about diffusion limits of Markov chain Monte Carlo
(MCMC) algorithms to the computer science notion of algorithm complexity. Our
main result states that any weak limit of a Markov process implies a
corresponding complexity bound (in an appropriate metric). We then combine this
result with previously-known MCMC diffusion limit results to prove that under
appropriate assumptions, the random-walk Metropolis algorithm in $d$ dimensions
takes $O(d)$ iterations to converge to stationarity, while the
Metropolis-adjusted Langevin algorithm takes $O(d^{1/3})$ iterations to
converge to stationarity.
</p>projecteuclid.org/euclid.jap/1466172863_20160617101421Fri, 17 Jun 2016 10:14 EDTStability of multiclass queueing networks under longest-queue and longest-dominating-queue schedulinghttp://projecteuclid.org/euclid.jap/1466172864<strong>Ramtin Pedarsani</strong>, <strong>Jean Walrand</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 421--433.</p><p><strong>Abstract:</strong><br/>
We consider the stability of robust scheduling policies for multiclass queueing
networks. These are open networks with arbitrary routeing matrix and several
disjoint groups of queues in which at most one queue can be served at a time.
The arrival and potential service processes and routeing decisions at the
queues are independent, stationary, and ergodic. A scheduling policy is called
\emph{robust} if it does not depend on the arrival and service rates nor on the
routeing probabilities. A policy is called \emph{throughput-optimal} if it
makes the system stable whenever the parameters are such that the system can be
stable. We propose two robust policies: longest-queue scheduling and a new
policy called longest-dominating-queue scheduling. We show that longest-queue
scheduling is throughput-optimal for two groups of two queues. We also prove
the throughput-optimality of longest-dominating-queue scheduling when the
network topology is acyclic, for an arbitrary number of groups and queues.
</p>projecteuclid.org/euclid.jap/1466172864_20160617101421Fri, 17 Jun 2016 10:14 EDTDegrees in random self-similar bipolar networkshttp://projecteuclid.org/euclid.jap/1466172865<strong>Chen Chen</strong>, <strong>Hosam Mahmoud</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 434--447.</p><p><strong>Abstract:</strong><br/>
We investigate several aspects of a self-similar evolutionary process that
builds a random bipolar network from building blocks that are themselves small
bipolar networks. We characterize admissible outdegrees in the history of the
evolution. We obtain the limit distribution of the polar degrees (when suitably
scaled) characterized by its sequence of moments. We also obtain the asymptotic
joint multivariate normal distribution of the number of nodes of small
admissible outdegrees. Five possible substructures arise, and each has its own
parameters (mean vector and covariance matrix) in the multivariate
distribution. Several results are obtained by mapping bipolar networks into
P\'olya urns.
</p>projecteuclid.org/euclid.jap/1466172865_20160617101421Fri, 17 Jun 2016 10:14 EDTRandom walks in a queueing network environmenthttp://projecteuclid.org/euclid.jap/1466172866<strong>M. Gannon</strong>, <strong>E. Pechersky</strong>, <strong> Y. Suhov</strong>, <strong>A. Yambartsev</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 448--462.</p><p><strong>Abstract:</strong><br/>
We propose a class of models of random walks in a random environment where an
exact solution can be given for a stationary distribution. The environment is
cast in terms of a Jackson/Gordon--Newell network although alternative
interpretations are possible. The main tool is the detailed balance equations.
The difference compared to earlier works is that the position of the random
walk influences the transition intensities of the network environment and vice
versa, creating strong correlations. The form of the stationary distribution is
closely related to the well-known product formula.
</p>projecteuclid.org/euclid.jap/1466172866_20160617101421Fri, 17 Jun 2016 10:14 EDTConvergence of directed random graphs to the Poisson-weighted infinite treehttp://projecteuclid.org/euclid.jap/1466172867<strong>Katja Gabrysch</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 463--474.</p><p><strong>Abstract:</strong><br/>
We consider a directed graph on the integers with a directed edge from vertex
$i$ to $j$ present with probability $n^{-1}$, whenever $i\lt j$, independently
of all other edges. Moreover, to each edge $(i,j)$ we assign weight
$n^{-1}(j-i)$. We show that the closure of vertex $0$ in such a weighted random
graph converges in distribution to the Poisson-weighted infinite tree as
$n\rightarrow \infty$. In addition, we derive limit theorems for the length of
the longest path in the subgraph of the Poisson-weighted infinite tree which
has all vertices at weighted distance of at most $\rho$ from the root.
</p>projecteuclid.org/euclid.jap/1466172867_20160617101421Fri, 17 Jun 2016 10:14 EDTRouteing on treeshttp://projecteuclid.org/euclid.jap/1466172868<strong>Maria Deijfen</strong>, <strong>Nina Gantert</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 475--488.</p><p><strong>Abstract:</strong><br/>
We consider three different schemes for signal routeing on a tree. The vertices
of the tree represent transceivers that can transmit and receive signals, and
are equipped with independent and identically distributed weights representing
the strength of the transceivers. The edges of the tree are also equipped with
independent and identically distributed weights, representing the costs for
passing the edges. For each one of our schemes, we derive sharp conditions on
the distributions of the vertex weights and the edge weights that determine
when the root can transmit a signal over arbitrarily large distances.
</p>projecteuclid.org/euclid.jap/1466172868_20160617101421Fri, 17 Jun 2016 10:14 EDTA household SIR epidemic model incorporating time of day effectshttp://projecteuclid.org/euclid.jap/1466172869<strong>Peter Neal</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 489--501.</p><p><strong>Abstract:</strong><br/>
During the course of a day an individual typically mixes with different groups
of individuals. Epidemic models incorporating population structure with
individuals being able to infect different groups of individuals have received
extensive attention in the literature. However, almost exclusively the models
assume that individuals are able to simultaneously infect members of all
groups, whereas in reality individuals will typically only be able to infect
members of any group they currently reside in. In this paper we develop a model
where individuals move between a community and their household during the
course of the day, only infecting within their current group. By defining a
novel branching process approximation with an explicit expression for the
probability generating function of the offspring distribution, we are able to
derive the probability of a major epidemic outbreak.
</p>projecteuclid.org/euclid.jap/1466172869_20160617101421Fri, 17 Jun 2016 10:14 EDTConvergence to the structured coalescent processhttp://projecteuclid.org/euclid.jap/1466172870<strong>Ryouta Kozakai</strong>, <strong>Akinobu Shimizu</strong>, <strong>Morihiro Notohara</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 502--517.</p><p><strong>Abstract:</strong><br/>
The coalescent was introduced by Kingman (1982a), (1982b) and Tajima (1983) as
a continuous-time Markov chain model describing the genealogical relationship
among sampled genes from a panmictic population of a species. The random mating
in a population is a strict condition and the genealogical structure of the
population has a strong influence on the genetic variability and the evolution
of the species. In this paper, starting from a discrete-time Markov chain
model, we show the weak convergence to a continuous-time Markov chain, called
the structured coalescent model, describing the genealogy of the sampled genes
from whole population by means of passing the limit of the population size.
Herbots (1997) proved the weak convergence to the structured coalescent on the
condition of conservative migration and Wright--Fisher-type reproduction. We
will give the proof on the condition of general migration rates and
exchangeable reproduction.
</p>projecteuclid.org/euclid.jap/1466172870_20160617101421Fri, 17 Jun 2016 10:14 EDTRespondent-driven sampling and an unusual epidemichttp://projecteuclid.org/euclid.jap/1466172871<strong>J. Malmros</strong>, <strong>F. Liljeros</strong>, <strong>T. Britton</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 518--530.</p><p><strong>Abstract:</strong><br/>
Respondent-driven sampling (RDS) is frequently used when sampling from hidden
populations. In RDS, sampled individuals pass on participation coupons to at
most $c$ of their acquaintances in the community ($c=3$ being a common choice).
If these individuals choose to participate, they in turn pass coupons on to
their acquaintances, and so on. The process of recruiting is shown to behave
like a new Reed--Frost-type network epidemic, in which `becoming infected'
corresponds to study participation. We calculate $R_0$, the probability of a
major `outbreak', and the relative size of a major outbreak for $c\lt\infty$ in
the limit of infinite population size and compare to the standard Reed--Frost
epidemic. Our results indicate that $c$ should often be chosen larger than in
current practice.
</p>projecteuclid.org/euclid.jap/1466172871_20160617101421Fri, 17 Jun 2016 10:14 EDTOptimal importance sampling for the Laplace transform of exponential Brownian functionalshttp://projecteuclid.org/euclid.jap/1466172872<strong>Je Guk Kim</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 531--542.</p><p><strong>Abstract:</strong><br/>
We present an asymptotically optimal importance sampling for Monte Carlo
simulation of the Laplace transform of exponential Brownian functionals which
plays a prominent role in many disciplines. To this end we utilize the theory
of large deviations to reduce finding an asymptotically optimal importance
sampling measure to solving a calculus of variations problem. Closed-form
solutions are obtained. In addition we also present a path to the test of
regularity of optimal drift which is an issue in implementing the proposed
method. The performance analysis of the method is provided through the Dothan
bond pricing model.
</p>projecteuclid.org/euclid.jap/1466172872_20160617101421Fri, 17 Jun 2016 10:14 EDTBoundary crossing probabilities for high-dimensional Brownian motionhttp://projecteuclid.org/euclid.jap/1466172873<strong>James C. Fu</strong>, <strong>Tung-Lung Wu</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 543--553.</p><p><strong>Abstract:</strong><br/>
The two-sided nonlinear boundary crossing probabilities for one-dimensional
Brownian motion and related processes have been studied in Fu and Wu (2010)
based on the finite Markov chain imbedding technique. It provides an efficient
numerical method to computing the boundary crossing probabilities. In this
paper we extend the above results for high-dimensional Brownian motion. In
particular, we obtain the rate of convergence for high-dimensional boundary
crossing probabilities. Numerical results are also provided to illustrate our
results.
</p>projecteuclid.org/euclid.jap/1466172873_20160617101421Fri, 17 Jun 2016 10:14 EDTOptimal dividend and reinsurance in the presence of two reinsurershttp://projecteuclid.org/euclid.jap/1466172874<strong>Mi Chen</strong>, <strong>Kam Chuen Yuen</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 554--571.</p><p><strong>Abstract:</strong><br/>
In this paper the optimal dividend (subject to transaction costs) and
reinsurance (with two reinsurers) problem is studied in the limit diffusion
setting. It is assumed that transaction costs and taxes are required when
dividends occur, and that the premiums charged by two reinsurers are calculated
according to the exponential premium principle with different parameters, which
makes the stochastic control problem nonlinear. The objective of the insurer is
to determine the optimal reinsurance and dividend policy so as to maximize the
expected discounted dividends until ruin. The problem is formulated as a mixed
classical-impulse stochastic control problem. Explicit expressions for the
value function and the corresponding optimal strategy are obtained. Finally, a
numerical example is presented to illustrate the impact of the parameters
associated with the two reinsurers' premium principle on the optimal
reinsurance strategy.
</p>projecteuclid.org/euclid.jap/1466172874_20160617101421Fri, 17 Jun 2016 10:14 EDTGerber–Shiu distribution at Parisian ruin for Lévy insurance risk processeshttp://projecteuclid.org/euclid.jap/1466172875<strong>Erik J. Baurdoux</strong>, <strong>Juan Carlos Pardo</strong>, <strong>José Luis Pérez</strong>, <strong>Jean-François Renaud</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 572--584.</p><p><strong>Abstract:</strong><br/>
Inspired by the works of Landriault \emph{et al.} (2011), (2014), we study the
Gerber--Shiu distribution at Parisian ruin with exponential implementation
delays for a spectrally negative L\'evy insurance risk process. To be more
specific, we study the so-called Gerber--Shiu distribution for a ruin model
where at each time the surplus process goes negative, an independent
exponential clock is started. If the clock rings before the surplus becomes
positive again then the insurance company is ruined. Our methodology uses
excursion theory for spectrally negative L\'evy processes and relies on the
theory of so-called scale functions. In particular, we extend the recent
results of Landriault \emph{et al.} (2011), (2014).
</p>projecteuclid.org/euclid.jap/1466172875_20160617101421Fri, 17 Jun 2016 10:14 EDTConvergence of tandem Brownian queueshttp://projecteuclid.org/euclid.jap/1466172876<strong>Sergio I. López</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 585--592.</p><p><strong>Abstract:</strong><br/>
It is known that in a stationary Brownian queue with both arrival and service
processes equal in law to Brownian motion, the departure process is a Brownian
motion, identical in law to the arrival process: this is the analogue of
Burke's theorem in this context. In this paper we prove convergence in law to
this Brownian motion in a tandem network of Brownian queues: if we have an
arbitrary continuous process, satisfying some mild conditions, as an initial
arrival process and pass it through an infinite tandem network of queues, the
resulting process weakly converges to a Brownian motion. We assume independent
and exponential initial workloads for all queues.
</p>projecteuclid.org/euclid.jap/1466172876_20160617101421Fri, 17 Jun 2016 10:14 EDTOn the invariance principle for reversible Markov chainshttp://projecteuclid.org/euclid.jap/1466172877<strong>Magda Peligrad</strong>, <strong>Sergey Utev</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 593--599.</p><p><strong>Abstract:</strong><br/>
In this paper we investigate the functional central limit theorem (CLT) for
stochastic processes associated to partial sums of additive functionals of
reversible Markov chains with general spate space, under the normalization
standard deviation of partial sums. For this case, we show that the functional
CLT is equivalent to the fact that the variance of partial sums is regularly
varying with exponent 1 and the partial sums satisfy the CLT. It is also
equivalent to the conditional CLT.
</p>projecteuclid.org/euclid.jap/1466172877_20160617101421Fri, 17 Jun 2016 10:14 EDTSparre Andersen identity and the last passage timehttp://projecteuclid.org/euclid.jap/1466172878<strong>Jevgenijs Ivanovs</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 600--605.</p><p><strong>Abstract:</strong><br/>
It is shown that the celebrated result of Sparre Andersen for random walks and
L\'{e}vy processes has intriguing consequences when the last time of the
process in $(-\infty,0]$, say $\sigma$, is added to the picture. In the case of
no positive jumps this leads to six random times, all of which have the same
distribution---the uniform distribution on $[0,\sigma]$. Surprisingly, this
result does not appear in the literature, even though it is based on some
classical observations concerning exchangeable increments.
</p>projecteuclid.org/euclid.jap/1466172878_20160617101421Fri, 17 Jun 2016 10:14 EDTDistribution of the quasispecies for a Galton–Watson process on the sharp peak landscapehttp://projecteuclid.org/euclid.jap/1466172879<strong>Joseba Dalmau</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 606--613.</p><p><strong>Abstract:</strong><br/>
We study a classical multitype Galton--Watson process with mutation and
selection. The individuals are sequences of fixed length over a finite
alphabet. On the sharp peak fitness landscape together with independent
mutations per locus, we show that, as the length of the sequences goes to
$\infty$ and the mutation probability goes to 0, the asymptotic relative
frequency of the sequences differing on $k$ digits from the master sequence
approaches $(\sigma\re^{-a}-1)({a^k}/{k!})\sum_{i\geq1}{i^k}/{\sigma^i},$ where
$\sigma$ is the selective advantage of the master sequence and $a$ is the
product of the length of the chains with the mutation probability. The
probability distribution $\mathcal{Q}(\sigma,a)$ on the nonnegative integers
given by the above equation is the quasispecies distribution with parameters
$\sigma$ and $a$.
</p>projecteuclid.org/euclid.jap/1466172879_20160617101421Fri, 17 Jun 2016 10:14 EDTA Galton–Watson process with a thresholdhttp://projecteuclid.org/euclid.jap/1466172880<strong>K. B. Athreya</strong>, <strong>H.-J. Schuh</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 614--621.</p><p><strong>Abstract:</strong><br/>
In this paper we study a special class of size dependent branching processes.
We assume that for some positive integer $K$ as long as the population size
does not exceed level $K$, the process evolves as a discrete-time supercritical
branching process, and when the population size exceeds level $K$, it evolves
as a subcritical or critical branching process. It is shown that this process
does die out in finite time $T$. The question of when the mean value
$\mathbb{E}(T)$ is finite or infinite is also addressed.
</p>projecteuclid.org/euclid.jap/1466172880_20160617101421Fri, 17 Jun 2016 10:14 EDTOptimization results for a generalized coupon collector problemhttp://projecteuclid.org/euclid.jap/1466172881<strong>Emmanuelle Anceaume</strong>, <strong>Yann Busnel</strong>, <strong>Ernst Schulte-Geers</strong>, <strong>Bruno Sericola</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 622--629.</p><p><strong>Abstract:</strong><br/>
In this paper we study a generalized coupon collector problem, which consists
of analyzing the time needed to collect a given number of distinct coupons that
are drawn from a set of coupons with an arbitrary probability distribution. We
suppose that a special coupon called the null coupon can be drawn but never
belongs to any collection. In this context, we prove that the almost uniform
distribution, for which all the nonnull coupons have the same drawing
probability, is the distribution which stochastically minimizes the time needed
to collect a fixed number of distinct coupons. Moreover, we show that in a
given closed subset of probability distributions, the distribution with all its
entries, but one, equal to the smallest possible value is the one which
stochastically maximizes the time needed to collect a fixed number of distinct
coupons.
</p>projecteuclid.org/euclid.jap/1466172881_20160617101421Fri, 17 Jun 2016 10:14 EDTA note on a generalized Ehrenfest urn model: another look at the mean transition timeshttp://projecteuclid.org/euclid.jap/1466172882<strong>Eve D. Lathrop</strong>, <strong>Isaac H. Goldstein</strong>, <strong>Yung-Pin Chen</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 2, 630--632.</p><p><strong>Abstract:</strong><br/>
This note is motivated by Blom's work in 1989. We consider a generalized
Ehrenfest urn model in which a randomly-chosen ball has a positive probability
of moving from one urn to the other urn. We use recursion relations between the
mean transition times to derive formulas in terms of finite sums, which are
shown to be equivalent to the definite integrals obtained by Blom.
</p>projecteuclid.org/euclid.jap/1466172882_20160617101421Fri, 17 Jun 2016 10:14 EDTSemiparametric cross entropy for rare-event simulationhttp://projecteuclid.org/euclid.jap/1476370767<strong>Z. I. Botev</strong>, <strong>A. Ridder</strong>, <strong>L. Rojas-Nandayapa</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 633--649.</p><p><strong>Abstract:</strong><br/>
The cross entropy is a well-known adaptive importance sampling method which requires estimating an optimal importance sampling distribution within a parametric class. In this paper we analyze an alternative version of the cross entropy, where the importance sampling distribution is selected instead within a general semiparametric class of distributions. We show that the semiparametric cross entropy method delivers efficient estimators in a wide variety of rare-event problems. We illustrate the favourable performance of the method with numerical experiments.
</p>projecteuclid.org/euclid.jap/1476370767_20161013105933Thu, 13 Oct 2016 10:59 EDT>Bit flipping and time to recoverhttp://projecteuclid.org/euclid.jap/1476370768<strong>Anton Muratov</strong>, <strong>Sergei Zuyev</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 650--666.</p><p><strong>Abstract:</strong><br/>
We call `bits' a sequence of devices indexed by positive integers, where every device can be in two states: 0 (idle) and 1 (active). Start from the `ground state' of the system when all bits are in 0-state. In our first binary flipping (BF) model the evolution of the system behaves as follows. At each time step choose one bit from a given distribution P on the positive integers independently of anything else, then flip the state of this bit to the opposite state. In our second damaged bits (DB) model a `damaged' state is added: each selected idling bit changes to active, but selecting an active bit changes its state to damaged in which it then stays forever. In both models we analyse the recurrence of the system's ground state when no bits are active. We present sufficient conditions for both the BF and DB models to show recurrent or transient behaviour, depending on the properties of the distribution P . We provide a bound for fractional moments of the return time to the ground state for the BF model, and prove a central limit theorem for the number of active bits for both models. </p>projecteuclid.org/euclid.jap/1476370768_20161013105933Thu, 13 Oct 2016 10:59 EDTSensor allocation problems on the real linehttp://projecteuclid.org/euclid.jap/1476370769<strong>Evangelos Kranakis</strong>, <strong>Gennady Shaikhet</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 667--687.</p><p><strong>Abstract:</strong><br/>
A large number n of sensors (finite connected intervals) are placed randomly on the real line so that the distances between the consecutive midpoints are independent random variables with expectation inversely proportional to n . In this work we address two fundamental sensor allocation problems. The interference problem tries to reallocate the sensors from their initial positions to eliminate overlaps. The coverage problem, on the other hand, allows overlaps, but tries to eliminate uncovered spaces between the originally placed sensors. Both problems seek to minimize the total sensor movement while reaching their respective goals. Using tools from queueing theory, Skorokhod reflections, and weak convergence, we investigate asymptotic behaviour of optimal costs as n increases to ∞. The introduced methodology is then used to address a more complicated, modified coverage problem, in which the overlaps between any two sensors can not exceed a certain parameter.
</p>projecteuclid.org/euclid.jap/1476370769_20161013105933Thu, 13 Oct 2016 10:59 EDTErgodicity of age-dependent inventory control systemshttp://projecteuclid.org/euclid.jap/1476370770<strong>Fredrik Olsson</strong>, <strong>Tatyana S. Turova</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 688--699.</p><p><strong>Abstract:</strong><br/>
We consider continuous review inventory systems with general doubly stochastic Poisson demand. In this specific case the demand rate, experienced by the system, varies as a function of the age of the oldest unit in the system. It is known that the stationary distributions of the ages in such models often have a strikingly simple form. In particular, they exhibit a typical feature of a Poisson process: given the age of the oldest unit the remaining ages are uniform. The model we treat here generalizes some known inventory models dealing with partial backorders, perishable items, and emergency replenishment. We derive the limiting joint density of the ages of the units in the system by solving partial differential equations. We also answer the question of the uniqueness of the stationary distributions which was not addressed in the related literature.
</p>projecteuclid.org/euclid.jap/1476370770_20161013105933Thu, 13 Oct 2016 10:59 EDTCompound geometric approximation under a failure rate constrainthttp://projecteuclid.org/euclid.jap/1476370771<strong>Fraser Daly</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 700--714.</p><p><strong>Abstract:</strong><br/>
We consider compound geometric approximation for a nonnegative, integer-valued random variable W . The bound we give is straightforward but relies on having a lower bound on the failure rate of W . Applications are presented to M/G/1 queuing systems, for which we state explicit bounds in approximations for the number of customers in the system and the number of customers served during a busy period. Other applications are given to birth–death processes and Poisson processes.
</p>projecteuclid.org/euclid.jap/1476370771_20161013105933Thu, 13 Oct 2016 10:59 EDTQuantile sensitivity estimation for dependent sequenceshttp://projecteuclid.org/euclid.jap/1476370772<strong>Guangxin Jiang</strong>, <strong>Michael C. Fu</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 715--732.</p><p><strong>Abstract:</strong><br/>
In this paper we estimate quantile sensitivities for dependent sequences via infinitesimal perturbation analysis, and prove asymptotic unbiasedness, weak consistency, and a central limit theorem for the estimators under some mild conditions. Two common cases, the regenerative setting and ϕ-mixing, are analyzed further, and a new batched estimator is constructed based on regenerative cycles for regenerative processes. Two numerical examples, the G/G/1 queue and the Ornstein–Uhlenbeck process, are given to show the effectiveness of the estimator.
</p>projecteuclid.org/euclid.jap/1476370772_20161013105933Thu, 13 Oct 2016 10:59 EDTOne-component regular variation and graphical modeling of extremeshttp://projecteuclid.org/euclid.jap/1476370773<strong>Adrien Hitz</strong>, <strong>Robin Evans</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 733--746.</p><p><strong>Abstract:</strong><br/>
The problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.
</p>projecteuclid.org/euclid.jap/1476370773_20161013105933Thu, 13 Oct 2016 10:59 EDTLaplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trialshttp://projecteuclid.org/euclid.jap/1476370774<strong>Takis Konstantopoulos</strong>, <strong>Zhenxia Liu</strong>, <strong>Xiangfeng Yang</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 747--764.</p><p><strong>Abstract:</strong><br/>
The longest stretch L ( n ) of consecutive heads in n
independent and identically distributed coin tosses is seen from the prism
of large deviations. We first establish precise asymptotics for the moment
generating function of L ( n ) and then show that there are
precisely two large deviation principles, one concerning the behavior of
the distribution of L ( n ) near its nominal value
log 1∕ p n and one away from it. We discuss
applications to inference and to logarithmic asymptotics of functionals of
L ( n ).
</p>projecteuclid.org/euclid.jap/1476370774_20161013105933Thu, 13 Oct 2016 10:59 EDTRenewal approximation for the absorption time of a decreasing Markov chainhttp://projecteuclid.org/euclid.jap/1476370775<strong>Gerold Alsmeyer</strong>, <strong>Alexander Marynych</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 765--782.</p><p><strong>Abstract:</strong><br/>
We consider a Markov chain
( M n ) n ≥0 on the set
ℕ 0 of nonnegative integers which is eventually
decreasing, i.e.
ℙ{ M n +1 < M n | M n ≥ a }=1
for some a ∈ℕ and all n ≥0. We are
interested in the asymptotic behavior of the law of the stopping time
T = T ( a )≔inf{ k ∈ℕ 0 : M k < a }
under
ℙ n ≔ℙ (· | M 0 = n )
as n →∞. Assuming that the decrements of
( M n ) n ≥0
given M 0 = n possess a kind of stationarity
for large n , we derive sufficient conditions for the convergence in
the minimal L p -distance of
ℙ n ( T − a n )∕ b n ∈·)
to some nondegenerate, proper law and give an explicit form of the
constants a n and b n .
</p>projecteuclid.org/euclid.jap/1476370775_20161013105933Thu, 13 Oct 2016 10:59 EDT>The SEIS model, or, the contact process with a latent stagehttp://projecteuclid.org/euclid.jap/1476370776<strong>Eric Foxall</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 783--801.</p><p><strong>Abstract:</strong><br/>
The susceptible→exposed→infectious→susceptible (SEIS) model is well known in mathematical epidemiology as a model of infection in which there is a latent period between the moment of infection and the onset of infectiousness. The compartment model is well studied, but the corresponding particle system has so far received no attention. For the particle system model in one spatial dimension, we give upper and lower bounds on the critical values, prove convergence of critical values in the limit of small and large latent time, and identify a limiting process to which the SEIS model converges in the limit of large latent time.
</p>projecteuclid.org/euclid.jap/1476370776_20161013105933Thu, 13 Oct 2016 10:59 EDTCoalescence on critical and subcritical multitype branching processeshttp://projecteuclid.org/euclid.jap/1476370777<strong>Jyy-I Hong</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 802--817.</p><p><strong>Abstract:</strong><br/>
Consider a d -type ( d <∞) Galton–Watson branching process, conditioned on the event that there are at least k ≥2 individuals in the n th generation, pick k individuals at random from the n th generation and trace their lines of descent backward in time till they meet. In this paper, the limit behaviors of the distributions of the generation number of the most recent common ancestor of any k chosen individuals and of the whole population are studied for both critical and subcritical cases. Also, we investigate the limit distribution of the joint distribution of the generation number and their types.
</p>projecteuclid.org/euclid.jap/1476370777_20161013105933Thu, 13 Oct 2016 10:59 EDTOn a class of reflected AR(1) processeshttp://projecteuclid.org/euclid.jap/1476370778<strong>Onno Boxma</strong>, <strong>Michel Mandjes</strong>, <strong>Josh Reed</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 818--832.</p><p><strong>Abstract:</strong><br/>
In this paper we study a reflected AR(1) process, i.e. a process ( Z n ) n obeying the recursion Z n +1 = max{ aZ n + X n ,0}, with ( X n ) n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Z n (in terms of transforms) in case X n can be written as Y n − B n , with ( B n ) n being a sequence of independent random variables which are all Exp(λ) distributed, and ( Y n ) n i.i.d.; when | a |<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that ( B n ) n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein–Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.
</p>projecteuclid.org/euclid.jap/1476370778_20161013105933Thu, 13 Oct 2016 10:59 EDTBounded-hop percolation and wireless communicationhttp://projecteuclid.org/euclid.jap/1476370779<strong>Christian Hirsch</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 833--845.</p><p><strong>Abstract:</strong><br/>
Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process, we investigate the distribution of the minimum number of hops that are needed to reach some point of the base station process. In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the base station process as its intensity tends to 0. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the base station process in a given number of hops.
</p>projecteuclid.org/euclid.jap/1476370779_20161013105933Thu, 13 Oct 2016 10:59 EDTLongest paths in random Apollonian networks and largest r -ary subtrees of random d -ary recursive treeshttp://projecteuclid.org/euclid.jap/1476370780<strong>Andrea Collevecchio</strong>, <strong>Abbas Mehrabian</strong>, <strong>Nick Wormald</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 846--856.</p><p><strong>Abstract:</strong><br/>
Let r and d be positive integers with
r < d . Consider a random d -ary tree constructed
as follows. Start with a single vertex, and in each time-step choose a
uniformly random leaf and give it d newly created offspring. Let
𝒯 d , t be the tree produced after t
steps. We show that there exists a fixed δ<1 depending on
d and r such that almost surely for all large t , every
r -ary subtree of 𝒯 d , t has less than
t δ vertices. The proof involves analysis that also
yields a related result. Consider the following iterative construction of a
random planar triangulation. Start with a triangle embedded in the plane.
In each step, choose a bounded face uniformly at random, add a vertex
inside that face and join it to the vertices of the face. In this way, one
face is destroyed and three new faces are created. After t steps, we
obtain a random triangulated plane graph with t +3 vertices,
which is called a random Apollonian network. We prove that there exists a
fixed δ<1, such that eventually every path in this graph has
length less than t 𝛿 , which verifies a conjecture
of Cooper and Frieze (2015).
</p>projecteuclid.org/euclid.jap/1476370780_20161013105933Thu, 13 Oct 2016 10:59 EDT>Anisotropic scaling of the random grain model with application to network traffichttp://projecteuclid.org/euclid.jap/1476370781<strong>Vytautė Pilipauskaitė</strong>, <strong>Donatas Surgailis</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 857--879.</p><p><strong>Abstract:</strong><br/>
We obtain a complete description of anisotropic scaling limits of the random grain model on the plane with heavy-tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian, and ‘intermediate’ infinitely divisible random fields. The asymptotic form of the covariance function of the random grain model is obtained. Application to superimposed network traffic is included.
</p>projecteuclid.org/euclid.jap/1476370781_20161013105933Thu, 13 Oct 2016 10:59 EDTDistribution of the smallest visited point in a greedy walk on the linehttp://projecteuclid.org/euclid.jap/1476370782<strong>Katja Gabrysch</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 880--887.</p><p><strong>Abstract:</strong><br/>
We consider a greedy walk on a Poisson process on the real line. It is known that the walk does not visit all points of the process. In this paper we first obtain some useful independence properties associated with this process which enable us to compute the distribution of the sequence of indices of visited points. Given that the walk tends to +∞, we find the distribution of the number of visited points in the negative half-line, as well as the distribution of the time at which the walk achieves its minimum.
</p>projecteuclid.org/euclid.jap/1476370782_20161013105933Thu, 13 Oct 2016 10:59 EDTOn randomly spaced observations and continuous-time random walkshttp://projecteuclid.org/euclid.jap/1476370783<strong>Bojan Basrak</strong>, <strong>Drago Špoljarić</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 888--898.</p><p><strong>Abstract:</strong><br/>
We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy-tailed steps, the limiting behavior of extreme observations until a given time t tends to be rather involved. We describe the asymptotics and extend several partial results which appeared in this setting. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous-time random walks.
</p>projecteuclid.org/euclid.jap/1476370783_20161013105933Thu, 13 Oct 2016 10:59 EDTCentral limit theorems for a hypergeometric randomly reinforced urnhttp://projecteuclid.org/euclid.jap/1476370784<strong>Irene Crimaldi</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 899--913.</p><p><strong>Abstract:</strong><br/>
We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color, given the past, is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.
</p>projecteuclid.org/euclid.jap/1476370784_20161013105933Thu, 13 Oct 2016 10:59 EDTThe dominating colour of an infinite Pólya urn modelhttp://projecteuclid.org/euclid.jap/1476370785<strong>Erik Thörnblad</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 914--924.</p><p><strong>Abstract:</strong><br/>
We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n ≥1, choose a ball from the urn uniformly at random. With probability ½< p <1, return the ball to the urn along with another ball of the same colour. With probability 1− p , recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri (2015), (2016) and Thörnblad (2015). We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p , and removed with probability 1− p . Our results here generalise a classical result about the Pólya urn model (which corresponds to p =1).
</p>projecteuclid.org/euclid.jap/1476370785_20161013105933Thu, 13 Oct 2016 10:59 EDTThe entirely coupled region of supercritical contact processeshttp://projecteuclid.org/euclid.jap/1476370786<strong>Achillefs Tzioufas</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 925--929.</p><p><strong>Abstract:</strong><br/>
We consider translation-invariant, finite-range, supercritical contact processes. We show the existence of unbounded space-time cones within which the descendancy of the process from full occupancy may with positive probability be identical to that of the process from the single site at its apex. The proof comprises an argument that leans upon refinements of a successful coupling among these two processes, and is valid in d -dimensions.
</p>projecteuclid.org/euclid.jap/1476370786_20161013105933Thu, 13 Oct 2016 10:59 EDTOptimality results for coupon collectionhttp://projecteuclid.org/euclid.jap/1476370787<strong>Mark Brown</strong>, <strong>Sheldon M. Ross</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 930--937.</p><p><strong>Abstract:</strong><br/>
We consider the coupon collection problem, where each coupon is one of the types 1,…, s with probabilities given by a vector 𝒑. For specified numbers r 1 ,…, r s , we are interested in finding 𝒑 that minimizes the expected time to obtain at least r i type- i coupons for all i =1,…, s . For example, for s =2, r 1 =1, and r 2 = r , we show that p 1 =(log r −log(log r ))∕ r is close to optimal.
</p>projecteuclid.org/euclid.jap/1476370787_20161013105933Thu, 13 Oct 2016 10:59 EDTUniqueness of the null solution to a nonlinear partial differential equation satisfied by the explosion probability of a branching diffusionhttp://projecteuclid.org/euclid.jap/1476370788<strong>K. Bruce Erickson</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 938--945.</p><p><strong>Abstract:</strong><br/>
The explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.
</p>projecteuclid.org/euclid.jap/1476370788_20161013105933Thu, 13 Oct 2016 10:59 EDT>Computable bounds of an 𝓁²-spectral gap for discrete Markov chains with band transition matriceshttp://projecteuclid.org/euclid.jap/1476370789<strong>Loïc Hervé</strong>, <strong>James Ledoux</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 946--952.</p><p><strong>Abstract:</strong><br/>
We analyse the 𝓁²(𝜋)-convergence rate
of irreducible and aperiodic Markov chains with N -band transition
probability matrix P and with invariant distribution 𝜋. This
analysis is heavily based on two steps. First, the study of the essential
spectral radius
r ess ( P |𝓁²(𝜋) )
of P |𝓁²(𝜋)
derived from Hennion’s quasi-compactness criteria. Second, the
connection between the spectral gap property (SG 2 )
of P on 𝓁²(𝜋) and the
V -geometric ergodicity of P . Specifically, the
(SG 2 ) is shown to hold under the condition
α 0 ≔∑ m =− N N lim sup i →+∞ ( P ( i , i + m ) P * ( i + m , i )
1∕2 <1. Moreover,
r ess ( P |𝓁²(𝜋) ≤α 0 .
Effective bounds on the convergence rate can be provided from a truncation
procedure.
</p>projecteuclid.org/euclid.jap/1476370789_20161013105933Thu, 13 Oct 2016 10:59 EDTAn extension of a convergence theorem for Markov chains arising in population geneticshttp://projecteuclid.org/euclid.jap/1476370790<strong>Martin Möhle</strong>, <strong>Morihiro Notohara</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 3, 953--956.</p><p><strong>Abstract:</strong><br/>
We analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N -band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius r ess ( P |𝓁²(𝜋) ) of P |𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG 2 ) of P on 𝓁²(𝜋) and the V -geometric ergodicity of P . Specifically, the (SG 2 ) is shown to hold under the condition α 0 ≔∑ m =− N N lim sup i →+∞ ( P ( i , i + m ) P * ( i + m , i ) 1∕2 <1. Moreover, r ess ( P |𝓁²(𝜋) ≤α 0 . Effective bounds on the convergence rate can be provided from a truncation procedure.
</p>projecteuclid.org/euclid.jap/1476370790_20161013105933Thu, 13 Oct 2016 10:59 EDTSolving finite time horizon Dynkin games by optimal switchinghttp://projecteuclid.org/euclid.jap/1481132829<strong>Randall Martyr</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 957--973.</p><p><strong>Abstract:</strong><br/>
This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent results on norm estimates for doubly reflected backward stochastic differential equations. This theory is then demonstrated numerically for the special cases of cancellable call and put options in a Black‒Scholes market.
</p>projecteuclid.org/euclid.jap/1481132829_20161207124718Wed, 07 Dec 2016 12:47 ESTOrder statistics with memory: a model with reliability applicationshttp://projecteuclid.org/euclid.jap/1481132830<strong>Alexander Katzur</strong>, <strong>Udo Kamps</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 974--988.</p><p><strong>Abstract:</strong><br/>
An extended model of order statistics based on possibly different distributions is introduced and analyzed. In the interpretation of successive failure times in a 𝑘-out-of-𝑛 system, say, until each failure, the time periods under previous (increasing) loads exerted on the remaining components are recorded. Then the lifetime distribution of the system depends on the complete failure scheme. Thus, order statistics with memory provide an alternative to the use of sequential order statistics, which form a Markov chain. The quantities as well as their spacings, the interoccurrence times, can be compared by means of stochastic ordering.
</p>projecteuclid.org/euclid.jap/1481132830_20161207124718Wed, 07 Dec 2016 12:47 ESTOn the long-range dependence of fractional Poisson and negative binomial processeshttp://projecteuclid.org/euclid.jap/1481132831<strong>A. Maheshwari</strong>, <strong>P. Vellaisamy</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 989--1000.</p><p><strong>Abstract:</strong><br/>
We discuss the short-range dependence (SRD) property of the increments of the fractional Poisson process, called the fractional Poissonian noise. We also establish that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property. Our definitions of the SRD/LRD properties are similar to those for a stationary process and different from those recently used in Biard and Saussereau (2014).
</p>projecteuclid.org/euclid.jap/1481132831_20161207124718Wed, 07 Dec 2016 12:47 ESTThe Boolean model in the Shannon regime: three thresholds and related asymptoticshttp://projecteuclid.org/euclid.jap/1481132832<strong>Venkat Anantharam</strong>, <strong>François Baccelli</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1001--1018.</p><p><strong>Abstract:</strong><br/>
Consider a family of Boolean models, indexed by integers n ≥1. The n th model features a Poisson point process in ℝ n of intensity e { n ρ n } , and balls of independent and identically distributed radii distributed like X̅ n √ n . Assume that ρ n →ρ as n →∞, and that X̅ n satisfies a large deviations principle. We show that there then exist the three deterministic thresholds τ d , the degree threshold, τ p , the percolation probability threshold, and τ v , the volume fraction threshold, such that, asymptotically as n tends to ∞, we have the following features. (i) For ρ<τ d , almost every point is isolated, namely its ball intersects no other ball; (ii) for τ d <ρ<τ p , the mean number of balls intersected by a typical ball converges to ∞ and nevertheless there is no percolation; (iii) for τ p <ρ<τ v , the volume fraction is 0 and nevertheless percolation occurs; (iv) for τ d <ρ<τ v , the mean number of balls intersected by a typical ball converges to ∞ and nevertheless the volume fraction is 0; (v) for ρ>τ v , the whole space is covered. The analysis of this asymptotic regime is motivated by problems in information theory, but it could be of independent interest in stochastic geometry. The relations between these three thresholds and the Shannon‒Poltyrev threshold are discussed.
</p>projecteuclid.org/euclid.jap/1481132832_20161207124718Wed, 07 Dec 2016 12:47 ESTA stochastic two-stage innovation diffusion model on a latticehttp://projecteuclid.org/euclid.jap/1481132833<strong>Cristian F. Coletti</strong>, <strong>Karina B. E. de Oliveira</strong>, <strong>Pablo M. Rodriguez</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1019--1030.</p><p><strong>Abstract:</strong><br/>
We propose a stochastic model describing a process of awareness, evaluation, and decision making by agents on the d -dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0, 1, 2. In this model 0 stands for ignorants, 1 for aware, and 2 for adopters. Aware and adopters inform its nearest ignorant neighbors about a new product innovation at rate λ. At rate α an agent in aware state becomes an adopter due to the influence of adopters' neighbors. Finally, aware and adopters forget the information about the new product, thus becoming ignorant, at rate 1. Our purpose is to analyze the influence of the parameters on the qualitative behavior of the process. We obtain sufficient conditions under which the innovation diffusion (and adoption) either becomes extinct or propagates through the population with positive probability.
</p>projecteuclid.org/euclid.jap/1481132833_20161207124718Wed, 07 Dec 2016 12:47 ESTThe deterministic Kermack‒McKendrick model bounds the general stochastic epidemichttp://projecteuclid.org/euclid.jap/1481132834<strong>Robert R. Wilkinson</strong>, <strong>Frank G. Ball</strong>, <strong>Kieran J. Sharkey</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1031--1040.</p><p><strong>Abstract:</strong><br/>
We prove that, for Poisson transmission and recovery processes, the classic susceptible→infected→recovered (SIR) epidemic model of Kermack and McKendrick provides, for any given time t >0, a strict lower bound on the expected number of susceptibles and a strict upper bound on the expected number of recoveries in the general stochastic SIR epidemic. The proof is based on the recent message passing representation of SIR epidemics applied to a complete graph.
</p>projecteuclid.org/euclid.jap/1481132834_20161207124718Wed, 07 Dec 2016 12:47 ESTA sharp lower bound for choosing the maximum of an independent sequencehttp://projecteuclid.org/euclid.jap/1481132835<strong>Pieter C. Allaart</strong>, <strong>José A. Islas</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1041--1051.</p><p><strong>Abstract:</strong><br/>
In this paper we consider a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win probability. Precisely, if X 1 ,..., X n are independent random variables with known continuous distributions and V n ( X 1 ,...,X n ):=sup τ ℙ( X τ = M n ), where M n ≔max{ X 1 ,..., X n } and the supremum is over all stopping times adapted to X 1 ,..., X n then V n ( X 1 ,..., X n )≥(1-1/ n ) n -1 , and this bound is attained. The method of proof consists in reducing the problem to that of a sequence of random variables taking at most two possible values, and then applying Bruss' sum-the-odds theorem, Bruss (2000). In order to obtain a sharp bound for each n , we improve Bruss' lower bound, Bruss (2003), for the sum-the-odds problem.
</p>projecteuclid.org/euclid.jap/1481132835_20161207124718Wed, 07 Dec 2016 12:47 ESTSequential stochastic assignment problem with time-dependent random success rateshttp://projecteuclid.org/euclid.jap/1481132836<strong>Golshid Baharian</strong>, <strong>Arash Khatibi</strong>, <strong>Sheldon H. Jacobson</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1052--1063.</p><p><strong>Abstract:</strong><br/>
The sequential stochastic assignment problem (SSAP) allocates distinct workers with deterministic values to sequentially arriving tasks with stochastic parameters to maximize the expected total reward. In this paper we study an extension of the SSAP, in which the worker values are considered to be random variables, taking on new values upon each task arrival. Several SSAP models with different assumptions on the distribution of the worker values and closed-form expressions for optimal assignment policies are presented.
</p>projecteuclid.org/euclid.jap/1481132836_20161207124718Wed, 07 Dec 2016 12:47 ESTStability of the stochastic matching modelhttp://projecteuclid.org/euclid.jap/1481132837<strong>Jean Mairesse</strong>, <strong>Pascal Moyal</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1064--1077.</p><p><strong>Abstract:</strong><br/>
We introduce and study a new model that we call the matching model . Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be matched . There is a finite set of classes 𝒱 for the items, and the allowed matchings depend on the classes, according to a matching graph on 𝒱. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a matching policy . When the sequence of classes of the arriving items is independent and identically distributed, the sequence of buffer-content is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is nonbipartite.
</p>projecteuclid.org/euclid.jap/1481132837_20161207124718Wed, 07 Dec 2016 12:47 ESTDetailed computational analysis of queueing-time distributions of the BMAP/G/1 queue using rootshttp://projecteuclid.org/euclid.jap/1481132838<strong>Gagandeep Singh</strong>, <strong>U. C. Gupta</strong>, <strong>M. L. Chaudhry</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1078--1097.</p><p><strong>Abstract:</strong><br/>
In this paper we present closed-form expressions for the distribution of the virtual (actual) queueing time for the BMAP/R/1 and BMAP/D/1 queues, where `R' represents a class of distributions having rational Laplace‒Stieltjes transforms. The closed-form analysis is based on the roots of the underlying characteristic equation. Numerical aspects have been tested for a variety of arrival and service-time distributions and results are matched with those obtained using the matrix-analytic method (MAM). Further, a comparative study of computation time of the proposed method with the MAM has been carried out. Finally, we also present closed-form expressions for the distribution of the virtual (actual) system time. The proposed method is analytically quite simple and easy to implement.
</p>projecteuclid.org/euclid.jap/1481132838_20161207124718Wed, 07 Dec 2016 12:47 ESTSteady-state analysis of a multiclass MAP/PH/ c queue with acyclic PH retrialshttp://projecteuclid.org/euclid.jap/1481132839<strong>Tuǧrul Dayar</strong>, <strong>M. Can Orhan</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1098--1110.</p><p><strong>Abstract:</strong><br/>
A multiclass c -server retrial queueing system in which customers arrive according to a class-dependent Markovian arrival process (MAP) is considered. Service and retrial times follow class-dependent phase-type (PH) distributions with the further assumption that PH distributions of retrial times are acyclic. A necessary and sufficient condition for ergodicity is obtained from criteria based on drifts. The infinite state space of the model is truncated with an appropriately chosen Lyapunov function. The truncated model is described as a multidimensional Markov chain, and a Kronecker representation of its generator matrix is numerically analyzed.
</p>projecteuclid.org/euclid.jap/1481132839_20161207124718Wed, 07 Dec 2016 12:47 ESTUniversality of load balancing schemes on the diffusion scalehttp://projecteuclid.org/euclid.jap/1481132840<strong>Debankur Mukherjee</strong>, <strong>Sem C. Borst</strong>, <strong>Johan S. H. van Leeuwaarden</strong>, <strong>Philip A. Whiting</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1111--1124.</p><p><strong>Abstract:</strong><br/>
We consider a system of N parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among d randomly selected servers otherwise (1≤ d ≤ N ). This load balancing scheme subsumes the so-called join-the-idle queue policy ( d =1) and the celebrated join-the-shortest queue policy ( d = N ) as two crucial special cases. We develop a stochastic coupling construction to obtain the diffusion limit of the queue process in the Halfin‒Whitt heavy-traffic regime, and establish that it does not depend on the value of d , implying that assigning tasks to idle servers is sufficient for diffusion level optimality.
</p>projecteuclid.org/euclid.jap/1481132840_20161207124718Wed, 07 Dec 2016 12:47 ESTNonergodic Jackson networks with infinite supply–local stabilization and local equilibrium analysishttp://projecteuclid.org/euclid.jap/1481132841<strong>Jennifer Sommer</strong>, <strong>Hans Daduna</strong>, <strong>Bernd Heidergott</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1125--1142.</p><p><strong>Abstract:</strong><br/>
Classical Jackson networks are a well-established tool for the analysis of complex systems. In this paper we analyze Jackson networks with the additional features that (i) nodes may have an infinite supply of low priority work and (ii) nodes may be unstable in the sense that the queue length at these nodes grows beyond any bound. We provide the limiting distribution of the queue length distribution at stable nodes, which turns out to be of product form. A key step in establishing this result is the development of a new algorithm based on adjusted traffic equations for detecting unstable nodes. Our results complement the results known in the literature for the subcases of Jackson networks with either infinite supply nodes or unstable nodes by providing an analysis of the significantly more challenging case of networks with both types of nonstandard node present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures.
</p>projecteuclid.org/euclid.jap/1481132841_20161207124718Wed, 07 Dec 2016 12:47 ESTAsymptotic frequency of shapes in supercritical branching treeshttp://projecteuclid.org/euclid.jap/1481132842<strong>Giacomo Plazzotta</strong>, <strong>Caroline Colijn</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1143--1155.</p><p><strong>Abstract:</strong><br/>
The shapes of branching trees have been linked to disease transmission patterns. In this paper we use the general Crump‒Mode‒Jagers branching process to model an outbreak of an infectious disease under mild assumptions. Introducing a new class of characteristic functions, we are able to derive a formula for the limit of the frequency of the occurrences of a given shape in a general tree. The computational challenges concerning the evaluation of this formula are in part overcome using the jumping chronological contour process. We apply the formula to derive the limit of the frequency of cherries, pitchforks, and double cherries in the constant-rate birth‒death model, and the frequency of cherries under a nonconstant death rate.
</p>projecteuclid.org/euclid.jap/1481132842_20161207124718Wed, 07 Dec 2016 12:47 ESTOn a coalescence process and its branching genealogyhttp://projecteuclid.org/euclid.jap/1481132843<strong>Nicolas Grosjean</strong>, <strong>Thierry Huillet</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1156--1165.</p><p><strong>Abstract:</strong><br/>
We define and analyze a coalescent process as a recursive box-filling process whose genealogy is given by an ancestral time-reversed, time-inhomogeneous Bienyamé‒Galton‒Watson process. Special interest is on the expected size of a typical box and its probability of being empty. Special cases leading to exact asymptotic computations are investigated when the coalescing mechanisms are either linear fractional or quadratic.
</p>projecteuclid.org/euclid.jap/1481132843_20161207124718Wed, 07 Dec 2016 12:47 ESTDistributions of jumps in a continuous-state branching process with immigrationhttp://projecteuclid.org/euclid.jap/1481132844<strong>Xin He</strong>, <strong>Zenghu Li</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1166--1177.</p><p><strong>Abstract:</strong><br/>
We study the distributional properties of jumps in a continuous-state branching process with immigration. In particular, a representation is given for the distribution of the first jump time of the process with jump size in a given Borel set. From this result we derive a characterization for the distribution of the local maximal jump of the process. The equivalence of this distribution and the total Lévy measure is then studied. For the continuous-state branching process without immigration, we also study similar problems for its global maximal jump.
</p>projecteuclid.org/euclid.jap/1481132844_20161207124718Wed, 07 Dec 2016 12:47 ESTA central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walkhttp://projecteuclid.org/euclid.jap/1481132845<strong>Alexander Iksanov</strong>, <strong>Zakhar Kabluchko</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1178--1192.</p><p><strong>Abstract:</strong><br/>
Let ( W n (θ)) n ∈ℕ 0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W _ ∞ (θ) its limit. Assuming essentially that the martingale ( W n (2θ)) n ∈ℕ 0 is uniformly integrable and that var W 1 (θ) is finite, we prove a functional central limit theorem for the tail process ( W ∞ (θ)- W n + r (θ)) r ∈ℕ 0 and a law of the iterated logarithm for W ∞ (θ)- W n (θ) as n →∞.
</p>projecteuclid.org/euclid.jap/1481132845_20161207124718Wed, 07 Dec 2016 12:47 ESTOn the emergence of random initial conditions in fluid limitshttp://projecteuclid.org/euclid.jap/1481132846<strong>A. D. Barbour</strong>, <strong>P. Chigansky</strong>, <strong>F. C. Klebaner</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1193--1205.</p><p><strong>Abstract:</strong><br/>
In the paper we present a phenomenon occurring in population processes that start near 0 and have large carrying capacity. By the classical result of Kurtz (1970), such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to the carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to ∞, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition is related to the martingale limit of an associated linear birth-and-death process.
</p>projecteuclid.org/euclid.jap/1481132846_20161207124718Wed, 07 Dec 2016 12:47 ESTAsymptotics for randomly reinforced urns with random barriershttp://projecteuclid.org/euclid.jap/1481132847<strong>Patrizia Berti</strong>, <strong>Irene Crimaldi</strong>, <strong>Luca Pratelli</strong>, <strong>Pietro Rigo</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1206--1220.</p><p><strong>Abstract:</strong><br/>
An urn contains black and red balls. Let Z n be the proportion of black balls at time n and 0≤ L < U ≤1 random barriers. At each time n , a ball b n is drawn. If b n is black and Z n -1 < U , then b n is replaced together with a random number B n of black balls. If b n is red and Z n -1 > L , then b n is replaced together with a random number R n of red balls. Otherwise, no additional balls are added, and b n alone is replaced. In this paper we assume that R n = B n . Then, under mild conditions, it is shown that Z n → a.s. Z for some random variable Z , and D n ≔ √ n ( Z n - Z ) →𝒩(0,σ 2 ) conditionally almost surely (a.s.), where σ 2 is a certain random variance. Almost sure conditional convergence means that ℙ( D n ∈⋅|𝒢 n ) → w 𝒩(0,σ 2 ) a.s., where ℙ( D n ∈⋅|𝒢 n ) is a regular version of the conditional distribution of D n given the past 𝒢 n . Thus, in particular, one obtains D n →𝒩(0,σ 2 ) stably. It is also shown that L < Z < U a.s. and Z has nonatomic distribution.
</p>projecteuclid.org/euclid.jap/1481132847_20161207124718Wed, 07 Dec 2016 12:47 ESTSynchronization and fluctuation theorems for interacting Friedman urnshttp://projecteuclid.org/euclid.jap/1481132848<strong>Neeraja Sahasrabudhe</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1221--1239.</p><p><strong>Abstract:</strong><br/>
We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for fluctuations around the synchronization limit.
</p>projecteuclid.org/euclid.jap/1481132848_20161207124718Wed, 07 Dec 2016 12:47 ESTA generalization of the Mabinogion sheep problem of D. Williamshttp://projecteuclid.org/euclid.jap/1481132849<strong>Yi-Shen Lin</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1240--1256.</p><p><strong>Abstract:</strong><br/>
In his well-known textbook Probability with Martingales , David Williams (1991) introduces the Mabinogion sheep problem in which there is a magical flock of sheep, some black, some white. At each stage n =1,2,..., a sheep (chosen randomly from the entire flock, independently of previous events) bleats; if this bleating sheep is white, one black sheep (if any remain) instantly becomes white; if the bleating sheep is black, one white sheep (if any remain) instantly becomes black. No births or deaths occur. Suppose that one may remove any number of white sheep from the flock at (the end of) each stage n =0,1,.... The object is to maximize the expected final number of black sheep. By applying the martingale optimality principle, Williams showed that the problem is solvable and admits a simple nice solution. In this paper we consider a generalization of the Mabinogion sheep problem with two parameters 0≤ p , q ≤1, denoted M ( p , q ), in which at each stage, when the bleating sheep is white (black, respectively), a black (white, respectively) sheep (if any remain) instantly becomes white (black, respectively), with probability p ( q , respectively) and nothing changes with probability 1- p (1- q , respectively). Note that the original problem corresponds to ( p , q )=(1,1). Following Williams' approach, we solve the two cases ( p , q )=(1,1/2) and (1/2,1) which admit simple solutions.
</p>projecteuclid.org/euclid.jap/1481132849_20161207124718Wed, 07 Dec 2016 12:47 ESTAn alternative axiomatic characterisation of pricing operatorshttp://projecteuclid.org/euclid.jap/1481132850<strong>Stefan Kassberger</strong>, <strong>Thomas Liebmann</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1257--1264.</p><p><strong>Abstract:</strong><br/>
In the spirit of the axiomatic approach by Rogers (1998) we show the equivalence between a set of assumptions on the behaviour of prices and the existence of a representation of these prices as conditional expectations. We rely on only weak assumptions and avoid any a priori modelling of negligible events or of any market filtration. Rather, both endogenously emerge along with the representation as conditional expectations.
</p>projecteuclid.org/euclid.jap/1481132850_20161207124718Wed, 07 Dec 2016 12:47 ESTImproved Chen‒Stein bounds on the probability of a unionhttp://projecteuclid.org/euclid.jap/1481132851<strong>Sheldon M. Ross</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1265--1270.</p><p><strong>Abstract:</strong><br/>
We improve the Chen‒Stein bounds when applied to the probability of a union. When the probability is small, the improvement in the distance from the lower to the upper bound is roughly a factor of 2. Further improvements are determined when the events of the union are either negatively or positively dependent.
</p>projecteuclid.org/euclid.jap/1481132851_20161207124718Wed, 07 Dec 2016 12:47 ESTCorrection: Fractional Poisson process: long-range dependence and applications in ruin theoryhttp://projecteuclid.org/euclid.jap/1481132852<strong>R. Biard</strong>, <strong>B. Saussereau</strong>. <p><strong>Source: </strong>Journal of Applied Probability, Volume 53, Number 4, 1271--1272.</p>projecteuclid.org/euclid.jap/1481132852_20161207124718Wed, 07 Dec 2016 12:47 EST