Advances in Applied Probability Articles (Project Euclid)
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The latest articles from Advances in 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 10:16 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Excursion sets of three classes of stable random fields
http://projecteuclid.org/euclid.aap/1275055229
<strong>Robert J. Adler</strong>, <strong>Gennady Samorodnitsky</strong>, <strong>Jonathan E. Taylor</strong><p><strong>Source: </strong>Adv. in Appl. Probab., Volume 42, Number 2, 293--318.</p><p><strong>Abstract:</strong><br/>
Studying the geometry generated by Gaussian and Gaussian-related random fields
via their excursion sets is now a well-developed and well-understood subject.
The purely non-Gaussian scenario has, however, not been studied at all. In this
paper we look at three classes of stable random fields, and obtain asymptotic
formulae for the mean values of various geometric characteristics of their
excursion sets over high levels. While the formulae are asymptotic, they
contain enough information to show that not only do stable random fields
exhibit geometric behaviour very different from that of Gaussian fields, but
they also differ significantly among themselves.
</p>projecteuclid.org/euclid.aap/1275055229_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTOptimal double stopping of a Brownian bridgehttp://projecteuclid.org/euclid.aap/1449859807<strong>Erik J. Baurdoux</strong>, <strong>Nan Chen</strong>, <strong>Budhi A. Surya</strong>, <strong>Kazutoshi Yamazaki</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 47, Number 4, 1212--1234.</p><p><strong>Abstract:</strong><br/>
We study optimal double stopping problems driven by a Brownian bridge. The
objective is to maximize the expected spread between the payoffs achieved at
the two stopping times. We study several cases where the solutions can be
solved explicitly by strategies of a threshold type.
</p>projecteuclid.org/euclid.aap/1449859807_20151211134959Fri, 11 Dec 2015 13:49 ESTSecrecy coverage in two dimensionshttp://projecteuclid.org/euclid.aap/1457466152<strong>Amites Sarkar</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 1--12.</p><p><strong>Abstract:</strong><br/>
Working in the infinite plane R 2 , consider a Poisson process of black points with intensity 1, and an independent Poisson process of red points with intensity λ. We grow a disc around each black point until it hits the nearest red point, resulting in a random configuration A λ , which is the union of discs centered at the black points. Next, consider a fixed disc of area n in the plane. What is the probability p λ ( n ) that this disc is covered by A λ ? We prove that if λ 3 n log n = y then, for sufficiently large n , e -8π 2 y ≤ p λ ( n ) ≤ e -2π 2 y /3 . The proofs reveal a new and surprising phenomenon, namely, that the obstructions to coverage occur on a wide range of scales.
</p>projecteuclid.org/euclid.aap/1457466152_20160308144239Tue, 08 Mar 2016 14:42 ESTQueues with advanced reservations: an infinite-server proxy for the bookings diaryhttp://projecteuclid.org/euclid.aap/1457466153<strong>R. J. Maillardet</strong>, <strong>P. G. Taylor</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 13--31.</p><p><strong>Abstract:</strong><br/>
Queues with advanced reservations are endemic in the real world. In such a queue, the 'arrival' process is an incoming stream of customer 'booking requests', rather than actual customers requiring immediate service. We consider a model with a Poisson booking request process with rate λ. Associated with each request is a pair of independent random variables ( R i , S i ) constituting a request for service over a period S i , starting at a time R i into the future. Our interest is in the probability that a customer will be rejected due to capacity constraints. We present a simulation of a finite-capacity queue in which we record the proportion of rejected customers, and then move to an analysis of a queue with infinitely-many servers. Obviously no customers are rejected in the latter case. However, the event that the arrival of the extra customer will cause the number of customers in the queue to exceed C at some point during its service can be used as a proxy for the event that the customer would have been rejected in a system with finite capacity C . We start by calculating the transient and stationary distributions for some performance measures for the infinite-server queue. By observing that the stationary measure for the bookings diary (that is, the list of customers currently on hand, together with their start times and service times) is the same as the law for the entire sample path of an infinite server queue with a specified nonhomogenous Poisson input process, which we call the bookings queue , we are able to write down expressions for the abovementioned probability that, at some time during a requested service, the number of customers exceeds C . This measure serves as a bound for the probability that an incoming arrival would be refused admission in a system with C servers and, for a well-dimensioned system, it is to be hoped that it is a good approximation. We test the quality of this approximation by comparing our analytical results for the infinite-server case against simulation results for the finite-server case.
</p>projecteuclid.org/euclid.aap/1457466153_20160308144239Tue, 08 Mar 2016 14:42 ESTDuality and complete convergence for multi-type additive growth modelshttp://projecteuclid.org/euclid.aap/1457466154<strong>Eric Foxall</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 32--51.</p><p><strong>Abstract:</strong><br/>
We consider a class of multi-type particle systems whose structure is similar to that of a contact process and show that additivity is equivalent to the existence of a dual process, extending a result of Harris. We prove a necessary and sufficient condition for the model to preserve positive correlations. We then show that complete convergence on Z d holds for a large subclass of models including the two-stage contact process and a household model, and give examples.
</p>projecteuclid.org/euclid.aap/1457466154_20160308144239Tue, 08 Mar 2016 14:42 ESTOptimal stopping rule for the full-information duration problem with random horizonhttp://projecteuclid.org/euclid.aap/1457466155<strong>Mitsushi Tamaki</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 52--68.</p><p><strong>Abstract:</strong><br/>
The full-information duration problem with a random number N of objects is considered. These objects appear sequentially and their values X k are observed, where X k , independent of N , are independent and identically distributed random variables from a known continuous distribution. The objective of the problem is to find a stopping rule that maximizes the duration of holding a relative maximum (e.g. the k th object is a relative maximum if X k = max{ X 1 , X 2 , . . ., X k }). We assume that N is a random variable with a known upper bound n , so two models, Model 1 and Model 2, can be considered according to whether the planning horizon is N or n . The structure of the optimal rule, which depends on the prior distribution assumed on N , is examined. The monotone rule is defined and a necessary and sufficient condition for the optimal rule to be monotone is given for both models. Special attention is paid to the class of priors such that N / n converges, as n → ∞, to a random variable V m having density f V m ( v ) = m (1 - v ) m -1 , 0 ≤ v ≤ 1 for a positive integer m . An interesting feature is that the optimal duration (relative to n ) for Model 2 is just ( m + 1) times as large as that for Model 1 asymptotically.
</p>projecteuclid.org/euclid.aap/1457466155_20160308144239Tue, 08 Mar 2016 14:42 ESTTheory of segmented particle filtershttp://projecteuclid.org/euclid.aap/1457466156<strong>Hock Peng Chan</strong>, <strong>Chiang-Wee Heng</strong>, <strong>Ajay Jasra</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 69--87.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behavior of a new particle filter approach for the estimation of hidden Markov models. In particular, we develop an algorithm where the latent-state sequence is segmented into multiple shorter portions, with an estimation technique based upon a separate particle filter in each portion. The partitioning facilitates the use of parallel processing, which reduces the wall-clock computational time. Based upon this approach, we introduce new estimators of the latent states and likelihood which have similar or better variance properties compared to estimators derived from standard particle filters. We show that the likelihood function estimator is unbiased, and show asymptotic normality of the underlying estimators.
</p>projecteuclid.org/euclid.aap/1457466156_20160308144239Tue, 08 Mar 2016 14:42 ESTOn comparing coherent systems with heterogeneous componentshttp://projecteuclid.org/euclid.aap/1457466157<strong>Francisco J. Samaniego</strong>, <strong>Jorge Navarro</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 88--111.</p><p><strong>Abstract:</strong><br/>
In this paper we investigate different methods that may be used to compare coherent systems having heterogeneous components. We consider both the case of systems with independent components and the case of systems with dependent components. In the first case, the comparisons are based on the new concept of the survival signature due to Coolen and Coolen-Maturi (2012) which extends the well-known concept of system signatures to the case of components with lifetimes that need not be independent and identically distributed. In the second case, the comparisons are based on the concept of distortion functions. A graphical procedure (called an RR-plot) is proposed as an alternative to the analytical methods when there are two types of components.
</p>projecteuclid.org/euclid.aap/1457466157_20160308144239Tue, 08 Mar 2016 14:42 ESTOptimal learning with non-Gaussian rewardshttp://projecteuclid.org/euclid.aap/1457466158<strong>Zi Ding</strong>, <strong>Ilya O. Ryzhov</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 112--136.</p><p><strong>Abstract:</strong><br/>
We propose a novel theoretical characterization of the optimal 'Gittins index' policy in multi-armed bandit problems with non-Gaussian, infinitely divisible reward distributions. We first construct a continuous-time, conditional Lévy process which probabilistically interpolates the sequence of discrete-time rewards. When the rewards are Gaussian, this approach enables an easy connection to the convenient time-change properties of a Brownian motion. Although no such device is available in general for the non-Gaussian case, we use optimal stopping theory to characterize the value of the optimal policy as the solution to a free-boundary partial integro-differential equation (PIDE). We provide the free-boundary PIDE in explicit form under the specific settings of exponential and Poisson rewards. We also prove continuity and monotonicity properties of the Gittins index in these two problems, and discuss how the PIDE can be solved numerically to find the optimal index value of a given belief state.
</p>projecteuclid.org/euclid.aap/1457466158_20160308144239Tue, 08 Mar 2016 14:42 ESTSIR epidemics on a scale-free spatial nested modular networkhttp://projecteuclid.org/euclid.aap/1457466159<strong>Alberto Gandolfi</strong>, <strong>Lorenzo Cecconi</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 137--162.</p><p><strong>Abstract:</strong><br/>
We propose a class of random scale-free spatial networks with nested community structures called SHEM and analyze Reed-Frost epidemics with community related independent transmissions. We show that in a specific example of the SHEM the epidemic threshold may be trivial or not as a function of the relation among community sizes, distribution of the number of communities, and transmission rates.
</p>projecteuclid.org/euclid.aap/1457466159_20160308144239Tue, 08 Mar 2016 14:42 ESTThe degree profile and weight in Apollonian networks and k -treeshttp://projecteuclid.org/euclid.aap/1457466160<strong>Panpan Zhang</strong>, <strong>Hosam Mahmoud</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 163--175.</p><p><strong>Abstract:</strong><br/>
We investigate the degree profile and total weight in Apollonian networks. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a Mittag-Leffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi-Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k -trees, with minor modification in the proof methods, done mutatis mutandis.
</p>projecteuclid.org/euclid.aap/1457466160_20160308144239Tue, 08 Mar 2016 14:42 ESTCorrelation formulas for Markovian network processes in a random environmenthttp://projecteuclid.org/euclid.aap/1457466161<strong>Hans Daduna</strong>, <strong>Ryszard Szekli</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 176--198.</p><p><strong>Abstract:</strong><br/>
We consider Markov processes, which describe, e.g. queueing network processes, in a random environment which influences the network by determining random breakdown of nodes, and the necessity of repair thereafter. Starting from an explicit steady-state distribution of product form available in the literature, we note that this steady-state distribution does not provide information about the correlation structure in time and space (over nodes). We study this correlation structure via one-step correlations for the queueing-environment process. Although formulas for absolute values of these correlations are complicated, the differences of correlations of related networks are simple and have a nice structure. We therefore compare two networks in a random environment having the same invariant distribution, and focus on the time behaviour of the processes when in such a network the environment changes or the rules for travelling are perturbed. Evaluating the comparison formulas we compare spectral gaps and asymptotic variances of related processes.
</p>projecteuclid.org/euclid.aap/1457466161_20160308144239Tue, 08 Mar 2016 14:42 ESTA comparison of random walks in dependent random environmentshttp://projecteuclid.org/euclid.aap/1457466162<strong>Werner R. W. Scheinhardt</strong>, <strong>Dirk P. Kroese</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 199--214.</p><p><strong>Abstract:</strong><br/>
We provide exact computations for the drift of random walks in dependent random environments, including k -dependent and moving average environments. We show how the drift can be characterized and evaluated using Perron-Frobenius theory. Comparing random walks in various dependent environments, we demonstrate that their drifts can exhibit interesting behavior that depends significantly on the dependency structure of the random environment.
</p>projecteuclid.org/euclid.aap/1457466162_20160308144239Tue, 08 Mar 2016 14:42 ESTThe speed of a random walk excited by its recent historyhttp://projecteuclid.org/euclid.aap/1457466163<strong>Ross G. Pinsky</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 215--234.</p><p><strong>Abstract:</strong><br/>
Let N and M be positive integers satisfying 1≤ M ≤ N , and let 0< p 0 < p 1 < 1. Define a process {X n } n=0 ∞ on ℤ as follows. At each step, the process jumps either one step to the right or one step to the left, according to the following mechanism. For the first N steps, the process behaves like a random walk that jumps to the right with probability p 0 and to the left with probability 1- p 0 . At subsequent steps the jump mechanism is defined as follows: if at least M out of the N most recent jumps were to the right, then the probability of jumping to the right is p 1 ; however, if fewer than M out of the N most recent jumps were to the right then the probability of jumping to the right is p 0 . We calculate the speed of the process. Then we let N → ∞ and M / N → r ∈[0,1], and calculate the limiting speed. More generally, we consider the above questions for a random walk with a finite number l of threshold levels, ( M i , p i ) i =1 l , above the pre-threshold level p 0 , as well as for one model with l = N such thresholds.
</p>projecteuclid.org/euclid.aap/1457466163_20160308144239Tue, 08 Mar 2016 14:42 ESTMarkov-modulated Ornstein-Uhlenbeck processeshttp://projecteuclid.org/euclid.aap/1457466164<strong>G. Huang</strong>, <strong>H. M. Jansen</strong>, <strong>M. Mandjes</strong>, <strong>P. Spreij</strong>, <strong>K. De Turck</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 235--254.</p><p><strong>Abstract:</strong><br/>
In this paper we consider an Ornstein-Uhlenbeck (OU) process ( M ( t )) t ≥0 whose parameters are determined by an external Markov process ( X ( t )) t ≥0 on a finite state space {1, . . ., d }; this process is usually referred to as Markov-modulated Ornstein-Uhlenbeck . We use stochastic integration theory to determine explicit expressions for the mean and variance of M ( t ). Then we establish a system of partial differential equations (PDEs) for the Laplace transform of M ( t ) and the state X ( t ) of the background process, jointly for time epochs t = t 1 , . . ., t K . Then we use this PDE to set up a recursion that yields all moments of M ( t ) and its stationary counterpart; we also find an expression for the covariance between M ( t ) and M ( t + u ). We then establish a functional central limit theorem for M ( t ) for the situation that certain parameters of the underlying OU processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, specific scalings lead to drastically different limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple OU processes.
</p>projecteuclid.org/euclid.aap/1457466164_20160308144239Tue, 08 Mar 2016 14:42 ESTPerturbation analysis of inhomogeneous finite Markov chainshttp://projecteuclid.org/euclid.aap/1457466165<strong>Bernd Heidergott</strong>, <strong>Haralambie Leahu</strong>, <strong>Andreas Löpker</strong>, <strong>Georg Pflug</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 255--273.</p><p><strong>Abstract:</strong><br/>
In this paper we provide a perturbation analysis of finite time-inhomogeneous Markov processes. We derive closed-form representations for the derivative of the transition probability at time t , with t > 0. Elaborating on this result, we derive simple gradient estimators for transient performance characteristics either taken at some fixed point in time t , or for the integrated performance over a time interval [0 , t ]. Bounds for transient performance sensitivities are presented as well. Eventually, we identify a structural property of the derivative of the generator matrix of a Markov chain that leads to a significant simplification of the estimators.
</p>projecteuclid.org/euclid.aap/1457466165_20160308144239Tue, 08 Mar 2016 14:42 ESTJoint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in viewhttp://projecteuclid.org/euclid.aap/1457466166<strong>Hélène Guérin</strong>, <strong>Jean-François Renaud</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 274--297.</p><p><strong>Abstract:</strong><br/>
We study the distribution E x [exp(- q ∫ 0 t 1 ( a , b ) ( X s )d s ); X t ∈ d y ], where -∞ ≤ a < b < ∞, and where q , t > 0 and x ∈ R for a spectrally negative Lévy process X . More precisely, we identify the Laplace transform with respect to t of this measure in terms of the scale functions of the underlying process. Our results are then used to price step options and the particular case of an exponential spectrally negative Lévy jump-diffusion model is discussed.
</p>projecteuclid.org/euclid.aap/1457466166_20160308144239Tue, 08 Mar 2016 14:42 ESTIrreversible investment under Lévy uncertainty: an equation for the optimal boundaryhttp://projecteuclid.org/euclid.aap/1457466167<strong>Giorgio Ferrari</strong>, <strong>Paavo Salminen</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 1, 298--314.</p><p><strong>Abstract:</strong><br/>
We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we show that the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank-El Karoui representation problem. Such a relation and the Wiener-Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Lévy process hits any point in R with positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of Cobb-Douglas type and CES type. In the former case the function is separable and in the latter case nonseparable.
</p>projecteuclid.org/euclid.aap/1457466167_20160308144239Tue, 08 Mar 2016 14:42 ESTJackson networks in nonautonomous random environmentshttp://projecteuclid.org/euclid.aap/1465490750<strong>Ruslan Krenzler</strong>, <strong>Hans Daduna</strong>, <strong>Sonja Otten</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 315--331.</p><p><strong>Abstract:</strong><br/>
We investigate queueing networks in a random environment. The impact of the
evolving environment on the network is by changing service capacities
(upgrading and/or degrading, breakdown, repair) when the environment changes
its state. On the other side, customers departing from the network may enforce
the environment to jump immediately. This means that the environment is
nonautonomous and therefore results in a rather complex two-way interaction,
especially if the environment is not itself Markov. To react to the changes of
the capacities we implement randomised versions of the well-known deterministic
rerouteing schemes 'skipping' (jump-over protocol) and `reflection' (repeated
service, random direction). Our main result is an explicit expression for the
joint stationary distribution of the queue-lengths vector and the environment
which is of product form.
</p>projecteuclid.org/euclid.aap/1465490750_20160609124600Thu, 09 Jun 2016 12:46 EDTOn the equivalence of systems of different sizes, with applications to system comparisonshttp://projecteuclid.org/euclid.aap/1465490751<strong>Bo H. Lindqvist</strong>, <strong>Francisco J. Samaniego</strong>, <strong>Arne B. Huseby</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 332--348.</p><p><strong>Abstract:</strong><br/>
The signature of a coherent system is a useful tool in the study and comparison
of lifetimes of engineered systems. In order to compare two systems of
different sizes with respect to their signatures, the smaller system needs to
be represented by an equivalent system of the same size as the larger system.
In the paper we show how to construct equivalent systems by adding irrelevant
components to the smaller system. This leads to simpler proofs of some current
key results, and throws new light on the interpretation of mixed systems. We
also present a sufficient condition for equivalence of systems of different
sizes when restricting to coherent systems. In cases where for a given system
there is no equivalent system of smaller size, we characterize the class of
lower-sized systems with a signature vector which stochastically dominates the
signature of the larger system. This setup is applied to an optimization
problem in reliability economics.
</p>projecteuclid.org/euclid.aap/1465490751_20160609124600Thu, 09 Jun 2016 12:46 EDTMarcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent datahttp://projecteuclid.org/euclid.aap/1465490752<strong>Michael A. Kouritzin</strong>, <strong>Samira Sadeghi</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 349--368.</p><p><strong>Abstract:</strong><br/>
The Marcinkiewicz strong law,
lim n →∞ (1 / n 1/ p )∑ k =1 n ( D k - D ) = 0
almost surely with p ∈ (1, 2), is studied for outer products
D k = { X k X ̅ k T },
where { X k } and { X ̅ k }
are both two-sided (multivariate) linear processes (with coefficient matrices
( C l ), ( C ̅ l ) and
independent and identically distributed zero-mean innovations {Ξ} and
{Ξ̅}). Matrix sequences C l and
C ̅ l can decay slowly enough (as
| l | → ∞) that
{ X k , X ̅ k } have
long-range dependence, while { D k } can have heavy
tails. In particular, the heavy-tail and long-range-dependence phenomena for
{ D k } are handled simultaneously and a new decoupling
property is proved that shows the convergence rate is determined by the worst
of the heavy tails or the long-range dependence, but not the combination. The
main result is applied to obtain a Marcinkiewicz strong law of large numbers
for stochastic approximation, nonlinear function forms, and autocovariances.
</p>projecteuclid.org/euclid.aap/1465490752_20160609124600Thu, 09 Jun 2016 12:46 EDTProbabilistic cellular automata with general alphabets possessing a Markov chain as an invariant distributionhttp://projecteuclid.org/euclid.aap/1465490753<strong>Jérôme Casse</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 369--391.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to probabilistic cellular automata (PCAs) on
N , Z or Z / n Z , depending on two neighbors
with a general alphabet E (finite or infinite, discrete or not). We
study the following question: under which conditions does a PCA possess a
Markov chain as an invariant distribution? Previous results in the literature
give some conditions on the transition matrix (for positive rate PCAs) when the
alphabet E is finite. Here we obtain conditions on the transition kernel
of a PCA with a general alphabet E . In particular, we show that the
existence of an invariant Markov chain is equivalent to the existence of a
solution to a cubic integral equation. One of the difficulties in passing from
a finite alphabet to a general alphabet comes from the problem of
measurability, and a large part of this work is devoted to clarifying these
issues.
</p>projecteuclid.org/euclid.aap/1465490753_20160609124600Thu, 09 Jun 2016 12:46 EDTMarkov decision process algorithms for wealth allocation problems with defaultable bondshttp://projecteuclid.org/euclid.aap/1465490754<strong>Iker Perez</strong>, <strong>David Hodge</strong>, <strong>Huiling Le</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 392--405.</p><p><strong>Abstract:</strong><br/>
In this paper we are concerned with analysing optimal wealth allocation
techniques within a defaultable financial market similar to Bielecki and Jang
(2007). We study a portfolio optimization problem combining a continuous-time
jump market and a defaultable security; and present numerical solutions through
the conversion into a Markov decision process and characterization of its value
function as a unique fixed point to a contracting operator. In this paper we
analyse allocation strategies under several families of utility functions, and
highlight significant portfolio selection differences with previously reported
results.
</p>projecteuclid.org/euclid.aap/1465490754_20160609124600Thu, 09 Jun 2016 12:46 EDTOptimal financing and dividend distribution in a general diffusion model with regime switchinghttp://projecteuclid.org/euclid.aap/1465490755<strong>Jinxia Zhu</strong>, <strong>Hailiang Yang</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 406--422.</p><p><strong>Abstract:</strong><br/>
We study the optimal financing and dividend distribution problem with
restricted dividend rates in a diffusion type surplus model, where the drift
and volatility coefficients are general functions of the level of surplus and
the external environment regime. The environment regime is modeled by a Markov
process. Both capital injection and dividend payments incur expenses. The
objective is to maximize the expectation of the total discounted dividends
minus the total cost of the capital injection. We prove that it is optimal to
inject capital only when the surplus tends to fall below 0 and to pay out
dividends at the maximal rate when the surplus is at or above the threshold,
dependent on the environment regime.
</p>projecteuclid.org/euclid.aap/1465490755_20160609124600Thu, 09 Jun 2016 12:46 EDTContinuous affine processes: transformations, Markov chains and life insurancehttp://projecteuclid.org/euclid.aap/1465490756<strong>Kristian Buchardt</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 423--442.</p><p><strong>Abstract:</strong><br/>
Affine processes possess the property that expectations of exponential affine
transformations are given by a set of Riccati differential equations, which is
the main feature of this popular class of processes. In this paper we
generalise these results for expectations of more general transformations. This
is of interest in, e.g. doubly stochastic Markov models, in particular in life
insurance. When using affine processes for modelling the transition rates and
interest rate, the results presented allow for easy calculation of transition
probabilities and expected present values.
</p>projecteuclid.org/euclid.aap/1465490756_20160609124600Thu, 09 Jun 2016 12:46 EDTOn a class of multivariate counting processeshttp://projecteuclid.org/euclid.aap/1465490757<strong>Ji Hwan Cha</strong>, <strong>Massimiliano Giorgio</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 443--462.</p><p><strong>Abstract:</strong><br/>
In this paper we define and study a new class of multivariate counting
processes, named `multivariate generalized Pólya process'. Initially, we
define and study the bivariate generalized Pólya process and briefly
discuss its reliability application. In order to derive the main properties of
the process, we suggest some key properties and an important characterization
of the process. Due to these properties and the characterization, the main
properties of the bivariate generalized Pólya process are obtained
efficiently. The marginal processes of the multivariate generalized
Pólya process are shown to be the univariate generalized Pólya
processes studied in Cha (2014). Given the history of a marginal process, the
conditional property of the other process is also discussed. The bivariate
generalized Pólya process is extended to the multivariate case. We
define a new dependence concept for multivariate point processes and, based on
it, we analyze the dependence structure of the multivariate generalized
Pólya process.
</p>projecteuclid.org/euclid.aap/1465490757_20160609124600Thu, 09 Jun 2016 12:46 EDTPopulation viewpoint on Hawkes processeshttp://projecteuclid.org/euclid.aap/1465490758<strong>Alexandre Boumezoued</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 463--480.</p><p><strong>Abstract:</strong><br/>
In this paper we focus on a class of linear Hawkes processes with general
immigrants. These are counting processes with shot-noise intensity, including
self-excited and externally excited patterns. For such processes, we introduce
the concept of the age pyramid which evolves according to immigration and
births. The virtue of this approach that combines an intensity process
definition and a branching representation is that the population age pyramid
keeps track of all past events. This is used to compute new distribution
properties for a class of Hawkes processes with general immigrants which
generalize the popular exponential fertility function. The pathwise
construction of the Hawkes process and its underlying population is also given.
</p>projecteuclid.org/euclid.aap/1465490758_20160609124600Thu, 09 Jun 2016 12:46 EDTPopulation models at stochastic timeshttp://projecteuclid.org/euclid.aap/1465490759<strong>Enzo Orsingher</strong>, <strong>Costantino Ricciuti</strong>, <strong>Bruno Toaldo</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 481--498.</p><p><strong>Abstract:</strong><br/>
In this paper we consider time-changed models of population evolution
X f ( t ) = X ( H f ( t )),
where X is a counting process and H f is a
subordinator with Laplace exponent f . In the case where X is a
pure birth process, we study the form of the distribution, the intertimes
between successive jumps, and the condition of explosion (also in the case of
killed subordinators). We also investigate the case where X represents a
death process (linear or sublinear) and study the extinction probabilities as a
function of the initial population size n 0 . Finally, the
subordinated linear birth–death process is considered. Special attention
is devoted to the case where birth and death rates coincide; the sojourn times
are also analysed.
</p>projecteuclid.org/euclid.aap/1465490759_20160609124600Thu, 09 Jun 2016 12:46 EDTCoupling on weighted branching treeshttp://projecteuclid.org/euclid.aap/1465490760<strong>Ningyuan Chen</strong>, <strong>Mariana Olvera-Cravioto</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 499--524.</p><p><strong>Abstract:</strong><br/>
In this paper we consider linear functions constructed on two different
weighted branching processes and provide explicit bounds for their
Kantorovich–Rubinstein distance in terms of couplings of their
corresponding generic branching vectors. Motivated by applications to the
analysis of random graphs, we also consider a variation of the weighted
branching process where the generic branching vector has a different dependence
structure from the usual one. By applying the bounds to sequences of weighted
branching processes, we derive sufficient conditions for the convergence in the
Kantorovich–Rubinstein distance of linear functions. We focus on the
case where the limits are endogenous fixed points of suitable smoothing
transformations.
</p>projecteuclid.org/euclid.aap/1465490760_20160609124600Thu, 09 Jun 2016 12:46 EDTShape theorems for Poisson hail on a bivariate groundhttp://projecteuclid.org/euclid.aap/1465490761<strong>François Baccelli</strong>, <strong>Héctor A. Chang-Lara</strong>, <strong>Sergey Foss</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 525--543.</p><p><strong>Abstract:</strong><br/>
We consider an extension of the Poisson hail model where the service speed is
either 0 or ∞ at each point of the Euclidean space. We use and develop
tools pertaining to sub-additive ergodic theory in order to establish shape
theorems for the growth of the ice-heap under light tail assumptions on the
hailstone characteristics. The asymptotic shape depends on the statistics of
the hailstones, the intensity of the underlying Poisson point process, and on
the geometrical properties of the zero speed set.
</p>projecteuclid.org/euclid.aap/1465490761_20160609124600Thu, 09 Jun 2016 12:46 EDTExtremes for the inradius in the Poisson line tessellationhttp://projecteuclid.org/euclid.aap/1465490762<strong>Nicolas Chenavier</strong>, <strong>Ross Hemsley</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 544--573.</p><p><strong>Abstract:</strong><br/>
A Poisson line tessellation is observed in the window
W ρ := B (0, π -1/2 ρ 1/2 )
for ρ > 0. With each cell of the tessellation, we associate the
inradius, which is the radius of the largest ball contained in the cell. Using
the Poisson approximation, we compute the limit distributions of the largest
and smallest order statistics for the inradii of all cells whose nuclei are
contained in W ρ as ρ goes to ∞.
We additionally prove that the limit shape of the cells minimising the inradius
is a triangle.
</p>projecteuclid.org/euclid.aap/1465490762_20160609124600Thu, 09 Jun 2016 12:46 EDTPlanar tessellations that have the half-Gilbert structurehttp://projecteuclid.org/euclid.aap/1465490763<strong>James Burridge</strong>, <strong>Richard Cowan</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 574--584.</p><p><strong>Abstract:</strong><br/>
In the full rectangular version of Gilbert's planar tessellation (see
Gilbert (1967), Mackisack and Miles (1996), and Burridge et al. (2013)),
lines extend either horizontally (with east- and west-growing rays) or
vertically (north- and south-growing rays) from seed points which form a
stationary Poisson point process, each ray stopping when it meets another ray
that has blocked its path. In the half-Gilbert rectangular version,
east- and south-growing rays, whilst having the potential to block each other,
do not interact with west and north rays, and vice versa. East- and
south-growing rays have a reciprocality of blocking , as do west and
north. In this paper we significantly expand and simplify the half-Gilbert
analytic results that we gave in Burridge et al. (2013). We also show
how the idea of reciprocality of blocking between rays can be used in a
much wider context, with rays not necessarily orthogonal and with seeds that
produce more than two rays.
</p>projecteuclid.org/euclid.aap/1465490763_20160609124600Thu, 09 Jun 2016 12:46 EDTOn a memory game and preferential attachment graphshttp://projecteuclid.org/euclid.aap/1465490764<strong>Hüseyin Acan</strong>, <strong>Paweł Hitczenko</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 585--609.</p><p><strong>Abstract:</strong><br/>
In their recent paper Velleman and Warrington (2013) analyzed the expected
values of some of the parameters in a memory game; namely, the length of the
game, the waiting time for the first match, and the number of lucky moves. In
this paper we continue this direction of investigation and obtain the limiting
distributions of those parameters. More specifically, we prove that when
suitably normalized, these quantities converge in distribution to a normal,
Rayleigh, and Poisson random variable, respectively. We also make a connection
between the memory game and one of the models of preferential attachment
graphs. In particular, as a by-product of our methods, we obtain the joint
asymptotic normality of the degree counts in the preferential attachment
graphs. Furthermore, we obtain simpler proofs (although without rate of
convergence) of some of the results of Peköz et al . (2014) on the
joint limiting distributions of the degrees of the first few vertices in
preferential attachment graphs. In order to prove that the length of the game
is asymptotically normal, our main technical tool is a limit result for the
joint distribution of the number of balls in a multitype generalized
Pólya urn model.
</p>projecteuclid.org/euclid.aap/1465490764_20160609124600Thu, 09 Jun 2016 12:46 EDTMaximizing the variance of the time to ruin in a multiplayer game with selectionhttp://projecteuclid.org/euclid.aap/1465490765<strong>Ilie Grigorescu</strong>, <strong>Yi-Ching Yao</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 2, 610--630.</p><p><strong>Abstract:</strong><br/>
We consider a game with K ≥ 2 players, each having an
integer-valued fortune. On each round, a pair ( i , j ) among the
players with nonzero fortunes is chosen to play and the winner is decided by
flipping a fair coin (independently of the process up to that time). The winner
then receives a unit from the loser. All other players' fortunes remain the
same. (Once a player's fortune reaches 0, this player is out of the game.) The
game continues until only one player wins all. The choices of pairs represent
the control present in the problem. While it is known that the expected time to
ruin (i.e. expected duration of the game) is independent of the choices of
pairs ( i , j ) (the strategies), our objective is to find a strategy
which maximizes the variance of the time to ruin. We show that the maximum
variance is uniquely attained by the (optimal) strategy, which always selects a
pair of players who have currently the largest fortunes. An explicit formula
for the maximum value function is derived. By constructing a simple martingale,
we also provide a short proof of a result of Ross (2009) that the expected time
to ruin is independent of the strategies. A brief discussion of the (open)
problem of minimizing the variance of the time to ruin is given.
</p>projecteuclid.org/euclid.aap/1465490765_20160609124600Thu, 09 Jun 2016 12:46 EDTCheeger inequalities for absorbing Markov chainshttp://projecteuclid.org/euclid.aap/1474296307<strong>Gary Froyland</strong>, <strong>Robyn M. Stuart</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 631--647.</p><p><strong>Abstract:</strong><br/>
We construct Cheeger-type bounds for the second eigenvalue of a substochastic transition probability matrix in terms of the Markov chain's conductance and metastability (and vice versa) with respect to its quasistationary distribution, extending classical results for stochastic transition matrices.
</p>projecteuclid.org/euclid.aap/1474296307_20160919104516Mon, 19 Sep 2016 10:45 EDTLarge deviations for the empirical measure of heavy-tailed Markov renewal processeshttp://projecteuclid.org/euclid.aap/1474296308<strong>Mauro Mariani</strong>, <strong>Lorenzo Zambotti</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 648--671.</p><p><strong>Abstract:</strong><br/>
A large deviations principle is established for the joint law of the empirical measure and the flow measure of a Markov renewal process on a finite graph. We do not assume any bound on the arrival times, allowing heavy-tailed distributions. In particular, the rate function is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behaviour highly different from what one may guess with a heuristic Donsker‒Varadhan analysis of the problem.
</p>projecteuclid.org/euclid.aap/1474296308_20160919104516Mon, 19 Sep 2016 10:45 EDTMarcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent datahttp://projecteuclid.org/euclid.aap/1474296309<strong>Hui He</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 672--690.</p><p><strong>Abstract:</strong><br/>
Given a supercritical Galton‒Watson process { Z n } and a positive sequence {ε n }, we study the limiting behaviors of ℙ( S Z n /Z n ≥ε n ) with sums S n of independent and identically distributed random variables X i and m =𝔼[ Z 1 ]. We assume that we are in the Schröder case with 𝔼 Z 1 log Z 1 <∞ and X 1 is in the domain of attraction of an α-stable law with 0<α<2. As a by-product, when Z 1 is subexponentially distributed, we further obtain the convergence rate of Z n +1 / Z n to m as n →∞.
</p>projecteuclid.org/euclid.aap/1474296309_20160919104516Mon, 19 Sep 2016 10:45 EDTMultivariate fractional Poisson processes and compound sumshttp://projecteuclid.org/euclid.aap/1474296310<strong>Luisa Beghin</strong>, <strong>Claudio Macci</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 691--711.</p><p><strong>Abstract:</strong><br/>
In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (nonfractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.
</p>projecteuclid.org/euclid.aap/1474296310_20160919104516Mon, 19 Sep 2016 10:45 EDTOn the capacity functional of excursion sets of Gaussian random fields on ℝ 2http://projecteuclid.org/euclid.aap/1474296311<strong>Marie Kratz</strong>, <strong>Werner Nagel</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 712--725.</p><p><strong>Abstract:</strong><br/>
When a random field ( X t , t ∈ℝ 2 ) is thresholded on a given level u , the excursion set is given by its indicator 1 [ u , ∞) ( X t ). The purpose of this work is to study functionals (as established in stochastic geometry) of these random excursion sets as, e.g. the capacity functional as well as the second moment measure of the boundary length. It extends results obtained for the one-dimensional case to the two-dimensional case, with tools borrowed from crossings theory, in particular, Rice methods, and from integral and stochastic geometry.
</p>projecteuclid.org/euclid.aap/1474296311_20160919104516Mon, 19 Sep 2016 10:45 EDTUrn sampling distributions giving alternate correspondences between two optimal stopping problemshttp://projecteuclid.org/euclid.aap/1474296312<strong>Mitsushi Tamaki</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 726--743.</p><p><strong>Abstract:</strong><br/>
The best-choice problem and the duration problem, known as versions of the secretary problem, are concerned with choosing an object from those that appear sequentially. Let ( B ,p) denote the best-choice problem and ( D ,p) the duration problem when the total number N of objects is a bounded random variable with prior p=( p 1 , p 2 ,..., p n ) for a known upper bound n . Gnedin (2005) discovered the correspondence relation between these two quite different optimal stopping problems. That is, for any given prior p, there exists another prior q such that ( D ,p) is equivalent to ( B ,q). In this paper, motivated by his discovery, we attempt to find the alternate correspondence {p ( m ) , m ≥0}, i.e. an infinite sequence of priors such that ( D ,p ( m -1) ) is equivalent to ( B ,p ( m ) ) for all m ≥1, starting with p (0) =(0,...,0,1). To be more precise, the duration problem is distinguished into ( D 1 ,p) or ( D 2 ,p), referred to as model 1 or model 2, depending on whether the planning horizon is N or n . The aforementioned problem is model 1. For model 2 as well, we can find the similar alternate correspondence {p [ m ] , m ≥ 0}. We treat both the no-information model and the full-information model and examine the limiting behaviors of their optimal rules and optimal values related to the alternate correspondences as n →∞. A generalization of the no-information model is given. It is worth mentioning that the alternate correspondences for model 1 and model 2 are respectively related to the urn sampling models without replacement and with replacement.
</p>projecteuclid.org/euclid.aap/1474296312_20160919104516Mon, 19 Sep 2016 10:45 EDTSeries expansions for the all-time maximum of α-stable random walkshttp://projecteuclid.org/euclid.aap/1474296313<strong>Clifford Hurvich</strong>, <strong>Josh Reed</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 744--767.</p><p><strong>Abstract:</strong><br/>
We study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.
</p>projecteuclid.org/euclid.aap/1474296313_20160919104516Mon, 19 Sep 2016 10:45 EDTSIR epidemics with stages of infectionhttp://projecteuclid.org/euclid.aap/1474296314<strong>Claude Lefèvre</strong>, <strong>Massimiliano Giorgio</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 768--791.</p><p><strong>Abstract:</strong><br/>
In this paper we are concerned with a stochastic model for the spread of an epidemic in a closed homogeneously mixing population when an infective can go through several stages of infection before being removed. The transitions between stages are governed by either a Markov process or a semi-Markov process. An infective of any stage makes contacts amongst the population at the points of a Poisson process. Our main purpose is to derive the distribution of the final epidemic size and severity, as well as an approximation by branching, using simple matrix analytic methods. Some illustrations are given, including a model with treatment discussed by Gani (2006).
</p>projecteuclid.org/euclid.aap/1474296314_20160919104516Mon, 19 Sep 2016 10:45 EDTAnalysis of a stochastic approximation algorithm for computing quasi-stationary distributionshttp://projecteuclid.org/euclid.aap/1474296315<strong>J. Blanchet</strong>, <strong>P. Glynn</strong>, <strong>S. Zheng</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 792--811.</p><p><strong>Abstract:</strong><br/>
We study the convergence properties of a Monte Carlo estimator proposed in the physics literature to compute the quasi-stationary distribution on a transient set of a Markov chain (see De Oliveira and Dickman (2005), (2006), and Dickman and Vidigal (2002)). Using the theory of stochastic approximations we verify the consistency of the estimator and obtain an associated central limit theorem. We provide an example showing that convergence might occur very slowly if a certain eigenvalue condition is violated. We alleviate this problem using an easy-to-implement projection step combined with averaging.
</p>projecteuclid.org/euclid.aap/1474296315_20160919104516Mon, 19 Sep 2016 10:45 EDTG/G/∞ queues with renewal alternating interruptionshttp://projecteuclid.org/euclid.aap/1474296316<strong>Guodong Pang</strong>, <strong>Yuhang Zhou</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 812--831.</p><p><strong>Abstract:</strong><br/>
We study G/G/∞ queues with renewal alternating service interruptions, where the service station experiences `up' and `down' periods. The system operates normally in the up periods, and all servers stop functioning while customers continue entering the system during the down periods. The amount of service a customer has received when an interruption occurs will be conserved and the service will resume when the down period ends. We use a two-parameter process to describe the system dynamics: X r ( t , y ) tracking the number of customers in the system at time t that have residual service times strictly greater than y . The service times are assumed to satisfy either of the two conditions: they are independent and identically distributed with a distribution of a finite support, or are a stationary and weakly dependent sequence satisfying the ϕ-mixing condition and having a continuous marginal distribution function. We consider the system in a heavy-traffic asymptotic regime where the arrival rate gets large and service time distribution is fixed, and the interruption down times are asymptotically negligible while the up times are of the same order as the service times. We show the functional law of large numbers and functional central limit theorem (FCLT) for the process X r ( t , y ) in this regime, where the convergence is in the space 𝔻([0,∞), (𝔻, L 1 )) endowed with the Skorokhod M 1 topology. The limit processes in the FCLT possess a stochastic decomposition property.
</p>projecteuclid.org/euclid.aap/1474296316_20160919104516Mon, 19 Sep 2016 10:45 EDTDynamic programming for discrete-time finite-horizon optimal switching problems with negative switching costshttp://projecteuclid.org/euclid.aap/1474296317<strong>R. Martyr</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 832--847.</p><p><strong>Abstract:</strong><br/>
In this paper we study a discrete-time optimal switching problem on a finite horizon. The underlying model has a running reward, terminal reward, and signed (positive and negative) switching costs. Using optimal stopping theory for discrete-parameter stochastic processes, we extend a well-known explicit dynamic programming method for computing the value function and the optimal strategy to the case of signed switching costs.
</p>projecteuclid.org/euclid.aap/1474296317_20160919104516Mon, 19 Sep 2016 10:45 EDTOn the relation between graph distance and Euclidean distance in random geometric graphshttp://projecteuclid.org/euclid.aap/1474296318<strong>J. Díaz</strong>, <strong>D. Mitsche</strong>, <strong>G. Perarnau</strong>, <strong>X. Pérez-Giménez</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 848--864.</p><p><strong>Abstract:</strong><br/>
Given any two vertices u , v of a random geometric graph G( n , r ), denote by d E ( u , v ) their Euclidean distance and by d E ( u , v ) their graph distance. The problem of finding upper bounds on d G ( u , v ) conditional on d E ( u , v ) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r =ω(√log n ) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on d E ( u , v ) conditional on d E ( u , v ).
</p>projecteuclid.org/euclid.aap/1474296318_20160919104516Mon, 19 Sep 2016 10:45 EDTDegrees and distances in random and evolving apollonian networkshttp://projecteuclid.org/euclid.aap/1474296319<strong>István Kolossváry</strong>, <strong>Júlia Komjáthy</strong>, <strong>Lajos Vágó</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 865--902.</p><p><strong>Abstract:</strong><br/>
In this paper we study random Apollonian networks (RANs) and evolving Apollonian networks (EANs), in d dimensions for any d ≥2, i.e. dynamically evolving random d -dimensional simplices, looked at as graphs inside an initial d -dimensional simplex. We determine the limiting degree distribution in RANs and show that it follows a power-law tail with exponent τ=(2 d -1)/( d -1). We further show that the degree distribution in EANs converges to the same degree distribution if the simplex-occupation parameter in the n th step of the dynamics tends to 0 but is not summable in n . This result gives a rigorous proof for the conjecture of Zhang et al. (2006) that EANs tend to exhibit similar behaviour as RANs once the occupation parameter tends to 0. We also determine the asymptotic behaviour of the shortest paths in RANs and EANs for any d ≥2. For RANs we show that the shortest path between two vertices chosen u.a.r. (typical distance), the flooding time of a vertex chosen uniformly at random, and the diameter of the graph after n steps all scale as a constant multiplied by log n . We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar central limit theorem for typical distances in EANs.
</p>projecteuclid.org/euclid.aap/1474296319_20160919104516Mon, 19 Sep 2016 10:45 EDTOn real growth and run-off companies in insurance ruin theoryhttp://projecteuclid.org/euclid.aap/1474296320<strong>Harri Nyrhinen</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 903--925.</p><p><strong>Abstract:</strong><br/>
We study solvency of insurers in a comprehensive model where various economic factors affect the capital developments of the companies. The main interest is in the impact of real growth to ruin probabilities. The volume of the business is allowed to increase or decrease. In the latter case, the study is focused on run-off companies. Our main results give sharp asymptotic estimates for infinite-time ruin probabilities.
</p>projecteuclid.org/euclid.aap/1474296320_20160919104516Mon, 19 Sep 2016 10:45 EDTRisk minimization for game options in markets imposing minimal transaction costshttp://projecteuclid.org/euclid.aap/1474296321<strong>Yan Dolinsky</strong>, <strong>Yuri Kifer</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 3, 926--946.</p><p><strong>Abstract:</strong><br/>
We study partial hedging for game options in markets with transaction costs bounded from below. More precisely, we assume that the investor's transaction costs for each trade are the maximum between proportional transaction costs and a fixed transaction cost. We prove that in the continuous-time Black‒Scholes (BS) model, there exists a trading strategy which minimizes the shortfall risk. Furthermore, we use binomial models in order to provide numerical schemes for the calculation of the shortfall risk and the corresponding optimal portfolio in the BS model.
</p>projecteuclid.org/euclid.aap/1474296321_20160919104516Mon, 19 Sep 2016 10:45 EDTRandom fields of bounded variation and computation of their variation intensityhttp://projecteuclid.org/euclid.aap/1482548424<strong>Galerne Bruno</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 947--971.</p><p><strong>Abstract:</strong><br/>
The main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called the variation intensity , is established. This expression shows, in particular, that the variation intensity only depends on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations. </p>projecteuclid.org/euclid.aap/1482548424_20161223220042Fri, 23 Dec 2016 22:00 ESTCriterion for unlimited growth of critical multidimensional stochastic modelshttp://projecteuclid.org/euclid.aap/1482548425<strong>Etienne Adam</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 972--988.</p><p><strong>Abstract:</strong><br/>
We give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton–Watson with immigration processes and also significantly improve some results on multitype size-dependent Galton–Watson processes. </p>projecteuclid.org/euclid.aap/1482548425_20161223220042Fri, 23 Dec 2016 22:00 ESTOn the evolution of topology in dynamic clique complexeshttp://projecteuclid.org/euclid.aap/1482548426<strong>Gugan C. Thoppe</strong>, <strong>D. Yogeshwaran</strong>, <strong>Robert J. Adler</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 989--1014.</p><p><strong>Abstract:</strong><br/>
We consider a time varying analogue of the Erdős–Rényi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve independently as continuous-time Markov chains. Our main result is that when the edge inclusion probability is of the form p = n α , where n is the number of vertices and α∈(-1/ k , -1/( k + 1)), then the process of the normalised k th Betti number of these dynamic clique complexes converges weakly to the Ornstein–Uhlenbeck process as n →∞. </p>projecteuclid.org/euclid.aap/1482548426_20161223220042Fri, 23 Dec 2016 22:00 ESTComparison of time-inhomogeneous Markov processeshttp://projecteuclid.org/euclid.aap/1482548427<strong>Rudger Rüschendorf</strong>, <strong>Alexander Schnurr</strong>, <strong>Victor Wolf</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1015--1044.</p><p><strong>Abstract:</strong><br/>
Comparison results are given for time-inhomogeneous Markov processes with respect to function classes with induced stochastic orderings. The main result states the comparison of two processes, provided that the comparability of their infinitesimal generators as well as an invariance property of one process is assumed. The corresponding proof is based on a representation result for the solutions of inhomogeneous evolution problems in Banach spaces, which extends previously known results from the literature. Based on this representation, an ordering result for Markov processes induced by bounded and unbounded function classes is established. We give various applications to time-inhomogeneous diffusions, to processes with independent increments, and to Lévy-driven diffusion processes.
</p>projecteuclid.org/euclid.aap/1482548427_20161223220042Fri, 23 Dec 2016 22:00 ESTFirst-passage times of two-dimensional Brownian motionhttp://projecteuclid.org/euclid.aap/1482548428<strong>Steven Kou</strong>, <strong>Haowen Zhong</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1045--1060.</p><p><strong>Abstract:</strong><br/>
First-passage times (FPTs) of two-dimensional Brownian motion have many applications in quantitative finance. However, despite various attempts since the 1960s, there are few analytical solutions available. By solving a nonhomogeneous modified Helmholtz equation in an infinite wedge, we find analytical solutions for the Laplace transforms of FPTs; these Laplace transforms can be inverted numerically. The FPT problems lead to a class of bivariate exponential distributions which are absolute continuous but do not have memoryless property. We also prove that the density of the absolute difference of FPTs tends to ∞ if and only if the correlation between the two Brownian motions is positive.
</p>projecteuclid.org/euclid.aap/1482548428_20161223220042Fri, 23 Dec 2016 22:00 ESTLarge deviation principles for connectable receivers in wireless networkshttp://projecteuclid.org/euclid.aap/1482548429<strong>Christian Hirsch</strong>, <strong>Benedikt Jahnel</strong>, <strong>Paul Keeler</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1061--1094.</p><p><strong>Abstract:</strong><br/>
We study large deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large deviation principle for the rescaled process of these receivers as the connection threshold tends to 0. Finally, we show how these results can be used to develop importance sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect. </p>projecteuclid.org/euclid.aap/1482548429_20161223220042Fri, 23 Dec 2016 22:00 ESTUniform approximation of the Cox–Ingersoll–Ross process via exact simulation at random timeshttp://projecteuclid.org/euclid.aap/1482548430<strong>Grigori N. Milstein</strong>, <strong>John Schoenmakers</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1095--1116.</p><p><strong>Abstract:</strong><br/>
In this paper we uniformly approximate the trajectories of the Cox–Ingersoll–Ross (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view, the proposed method gives a better quality of approximation in a path-wise sense than standard, or even exact, simulation of the CIR dynamics at some deterministic time grid.
</p>projecteuclid.org/euclid.aap/1482548430_20161223220042Fri, 23 Dec 2016 22:00 ESTNonparametric estimation of the service time distribution in the M/G/∞ queuehttp://projecteuclid.org/euclid.aap/1482548431<strong>Alexander Goldenschluger</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1117--1138.</p><p><strong>Abstract:</strong><br/>
The subject of this paper is the problem of estimating the service time distribution of the M/G/∞ queue from incomplete data on the queue. The goal is to estimate G from observations of the queue-length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. An upper bound on the maximal risk is derived. The problem of estimating the arrival rate is considered as well. </p>projecteuclid.org/euclid.aap/1482548431_20161223220042Fri, 23 Dec 2016 22:00 ESTA queueing/inventory and an insurance risk modelhttp://projecteuclid.org/euclid.aap/1482548432<strong>Onno Boxma</strong>, <strong>Rim Essifi</strong>, <strong>Augustus J. E. M. Janssen</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1139--1160.</p><p><strong>Abstract:</strong><br/>
We study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity ω( x ) when the inventory is at level x >0, the present inventory is removed, instantaneously reducing the inventory to 0. We study the steady-state distribution of the (positive and negative) workload levels for the cases ω( x ) is constant and ω( x ) = a x . The key tool is the Wiener–Hopf factorization technique. When ω( x ) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace–Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model arising from insurance risk theory. </p>projecteuclid.org/euclid.aap/1482548432_20161223220042Fri, 23 Dec 2016 22:00 ESTA stochastic differential game for quadratic-linear diffusion processeshttp://projecteuclid.org/euclid.aap/1482548433<strong>Shangzhen Luo</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1161--1182.</p><p><strong>Abstract:</strong><br/>
In this paper we study a stochastic differential game between two insurers whose surplus processes are modelled by quadratic-linear diffusion processes. We consider an exit probability game. One insurer controls its risk process to minimize the probability that the surplus difference reaches a low level (indicating a disadvantaged surplus position of the insurer) before reaching a high level, while the other insurer aims to maximize the probability. We solve the game by finding the value function and the Nash equilibrium strategy in explicit forms. </p>projecteuclid.org/euclid.aap/1482548433_20161223220042Fri, 23 Dec 2016 22:00 ESTSpeed of coming down from infinity for birth-and-death processeshttp://projecteuclid.org/euclid.aap/1482548434<strong>Vincent Bansaye</strong>, <strong>Sylvie Méléard</strong>, <strong>Richard Mathieu</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1183--1210.</p><p><strong>Abstract:</strong><br/>
We describe in detail the speed of `coming down from infinity' for birth-and-death processes which eventually become extinct. Under general assumptions on the birth-and-death rates, we firstly determine the behavior of the successive hitting times of large integers. We identify two different regimes depending on whether the mean time for the process to go from n +1 to n is negligible or not compared to the mean time to reach n from ∞. In the first regime, the coming down from infinity is very fast and the convergence is weak. In the second regime, the coming down from infinity is gradual and a law of large numbers and a central limit theorem for the hitting times sequence hold. By an inversion procedure, we deduce that the process is almost surely equivalent to a nonincreasing function when the time goes to 0. Our results are illustrated by several examples including applications to population dynamics and population genetics. The particular case where the death rate varies regularly is studied in detail. </p>projecteuclid.org/euclid.aap/1482548434_20161223220042Fri, 23 Dec 2016 22:00 ESTBranching processes in generalized autoregressive conditional environmentshttp://projecteuclid.org/euclid.aap/1482548435<strong>Irene Hueter</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1211--1234.</p><p><strong>Abstract:</strong><br/>
Branching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality. </p>projecteuclid.org/euclid.aap/1482548435_20161223220042Fri, 23 Dec 2016 22:00 ESTOn long-range dependence of random measureshttp://projecteuclid.org/euclid.aap/1482548436<strong>Daniel Vašata</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1235--1255.</p><p><strong>Abstract:</strong><br/>
This paper deals with long-range dependence of random measures on ℝ d . By examples, it is demonstrated that one must be careful in order to define it consistently. Therefore, we define long-range dependence by a rather specific second-order condition and provide an equivalent formulation involving the asymptotic behaviour of the Bartlett spectrum near the origin. Then it is shown that the defining condition may be formulated less strictly when the additional isotropy assumption holds. Finally, we present an example of a long-range dependent random measure based on the 0-level excursion set of a Gaussian random field for which the corresponding spectral density and its asymptotics are explicitly derived. </p>projecteuclid.org/euclid.aap/1482548436_20161223220042Fri, 23 Dec 2016 22:00 ESTThe limit distribution of the largest interpoint distance for distributions supported by a d -dimensional ellipsoid and generalizationshttp://projecteuclid.org/euclid.aap/1482548437<strong>Michael Schrempp</strong>. <p><strong>Source: </strong>Advances in Applied Probability, Volume 48, Number 4, 1256--1270.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behaviour of the maximum interpoint distance of random points in a d -dimensional ellipsoid with a unique major axis. Instead of investigating only a fixed number of n points as n tends to ∞, we consider the much more general setting in which the random points are the supports of appropriately defined Poisson processes. Our main result covers the case of uniformly distributed points. </p>projecteuclid.org/euclid.aap/1482548437_20161223220042Fri, 23 Dec 2016 22:00 EST