Journal of Applied Probability Articles (Project Euclid)

Efficient importance sampling in ruin problems for multidimensional regularly varying random walks

Thu, 05 Aug 2010 15:41 EDT

Jose Blanchet, Jingchen Liu

Source: J. Appl. Probab., Volume 47, Number 2, 301--322.

Abstract:
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.

Sample path optimal policies for serial lines with flexible workers

Sat, 16 Jun 2012 16:34 EDT

Dimitrios G. Pandelis, Mark P. Van Oyen

Source: J. Appl. Probab., Volume 49, Number 2, 582--589.

Abstract:
We study the dynamic assignment of cross-trained workers in serial production lines characterized by stochastic process times and inventory buffers between stations. Throughput maximization is the objective. Each worker is trained for a subset of tasks, where emphasis is placed on systems with each worker trained for a zone of stations with stations near the zone boundaries being served (shared) by one or more other workers as well. Using sample path comparisons, we identify structural properties of optimal worker allocation policies. We identify when (i) a worker can prioritize the job in the most downstream station (last-buffer-first-served), and (ii) only the downstream (as opposed to upstream) server should serve a single task.

Means and variances in stochastic multistage cancer models

Sat, 16 Jun 2012 16:34 EDT

Aidan Sudbury

Source: J. Appl. Probab., Volume 49, Number 2, 590--594.

Abstract:
A widely used model of carcinogenesis assumes that cells must go through a process of acquiring several mutations before they become cancerous. This implies that at any time there will be several populations of cells at different stages of mutation. In this paper we give exact expressions for the expectations and variances of the number of cells in each stage of such a stochastic multistage cancer model .

OBITUARY: Miloslav Jiřina

Sat, 16 Jun 2012 16:34 EDT

John Darroch, Eugene Seneta

Source: J. Appl. Probab., Volume 49, Number 2, 595--599.

Birth of a strongly connected giant in an inhomogeneous random digraph

Thu, 06 Sep 2012 14:16 EDT

Mindaugas Bloznelis, Friedrich Götze, Jerzy Jaworski

Source: J. Appl. Probab., Volume 49, Number 3, 601--611.

Abstract:
We present and investigate a general model for inhomogeneous random digraphs with labeled vertices, where the arcs are generated independently, and the probability of inserting an arc depends on the labels of its endpoints and on its orientation. For this model, the critical point for the emergence of a giant component is determined via a branching process approach.

Coagulation processes with Gibbsian time evolution

Thu, 06 Sep 2012 14:16 EDT

Boris L. Granovsky, Alexander V. Kryvoshaev

Source: J. Appl. Probab., Volume 49, Number 3, 612--626.

Abstract:
We prove that a stochastic process of pure coagulation has at any time t ≥ 0 a time-dependent Gibbs distribution if and only if the rates ψ( i , j ) of single coagulations are of the form ψ( i ; j ) = if ( j ) + jf ( i ), where f is an arbitrary nonnegative function on the set of positive integers. We also obtain a recurrence relation for weights of these Gibbs distributions that allow us to derive the general form of the solution and the explicit solutions in three particular cases of the function f . For the three corresponding models, we study the probability of coagulation into one giant cluster by time t > 0.

Coalescence in critical and subcritical Galton-Watson branching processes

Thu, 06 Sep 2012 14:16 EDT

K. B. Athreya

Source: J. Appl. Probab., Volume 49, Number 3, 627--638.

Abstract:
In a Galton-Watson branching process that is not extinct by the n th generation and has at least two individuals, pick two individuals at random by simple random sampling without replacement. Trace their lines of descent back in time till they meet. Call that generation X n a pairwise coalescence time . Similarly, let Y n denote the coalescence time for the whole population of the n th generation conditioned on the event that it is not extinct. In this paper the distributions of X n and Y n , and their limit behaviors as n → ∞ are discussed for both the critical and subcritical cases.

Extinction probabilities of supercritical decomposable branching processes

Thu, 06 Sep 2012 14:16 EDT

Sophie Hautphenne

Source: J. Appl. Probab., Volume 49, Number 3, 639--651.

Abstract:
We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

Tightness for maxima of generalized branching random walks

Thu, 06 Sep 2012 14:16 EDT

Ming Fang

Source: J. Appl. Probab., Volume 49, Number 3, 652--670.

Abstract:
We study generalized branching random walks on the real line R that allow time dependence and local dependence between siblings. Specifically, starting from one particle at time 0, the system evolves such that each particle lives for one unit amount of time, gives birth independently to a random number of offspring according to some branching law, and dies. The offspring from a single particle are assumed to move to new locations on R according to some joint displacement distribution; the branching laws and displacement distributions depend on time. At time n , F n (·) is used to denote the distribution function of the position of the rightmost particle in generation n . Under appropriate tail assumptions on the branching laws and offspring displacement distributions, we prove that F n (· - Med( F n )) is tight in n , where Med( F n ) is the median of F n . The main part of the argument is to demonstrate the exponential decay of the right tail 1 - F n (· - Med( F n )).

An application of the backbone decomposition to supercritical super-Brownian motion with a barrier

Thu, 06 Sep 2012 14:16 EDT

A. E. Kyprianou, A. Murillo-Salas, J. L. Pérez

Source: J. Appl. Probab., Volume 49, Number 3, 671--684.

Abstract:
We analyse the behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki, Kyprianou and Murillo-Salas (2011). In particular, by considering existing results for branching Brownian motion due to Harris and Kyprianou (2006) and Maillard (2011), we obtain, with relative ease, conclusions regarding the growth in the right-most point in the support, analytical properties of the associated one-sided Fisher-Kolmogorov-Petrovskii-Piscounov wave equation, as well as the distribution of mass on the exit measure associated with the barrier.

First passage time of skew Brownian motion

Thu, 06 Sep 2012 14:16 EDT

Thilanka Appuhamillage, Daniel Sheldon

Source: J. Appl. Probab., Volume 49, Number 3, 685--696.

Abstract:
Nearly fifty years after the introduction of skew Brownian motion by Itô and McKean (1963), the first passage time distribution remains unknown. In this paper we first generalize results of Pitman and Yor (2011) and Csáki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion, and then use these results to derive the first passage time distribution.

Exit problems for reflected Markov-modulated Brownian motion

Thu, 06 Sep 2012 14:16 EDT

Lothar Breuer

Source: J. Appl. Probab., Volume 49, Number 3, 697--709.

Abstract:
Let ( X , J ) denote a Markov-modulated Brownian motion (MMBM) and denote its supremum process by S . For some a > 0, let σ( a ) denote the time when the reflected process Y := S - X first surpasses the level a . Furthermore, let σ - ( a ) denote the last time before σ( a ) when X attains its current supremum. In this paper we shall derive the joint distribution of S σ( a ) , σ - ( a ), and σ( a ), where the latter two will be given in terms of their Laplace transforms. We also provide some remarks on scale matrices for MMBMs with strictly positive variation parameters. This extends recent results for spectrally negative Lévy processes to MMBMs. Due to well-known fluid embedding and state-dependent killing techniques, the analysis applies to Markov additive processes with phase-type jumps as well. The result is of interest to applications such as the dividend problem in insurance mathematics and the buffer overflow problem in queueing theory. Examples will be given for the former.

Fractional Brownian motion with H < 1/2 as a limit of scheduled traffic

Thu, 06 Sep 2012 14:16 EDT

Victor F. Araman, Peter W. Glynn

Source: J. Appl. Probab., Volume 49, Number 3, 710--718.

Abstract:
In this paper we show that fractional Brownian motion with H < ½ can arise as a limit of a simple class of traffic processes that we call 'scheduled traffic models'. To our knowledge, this paper provides the first simple traffic model leading to fractional Brownnian motion with H < ½. We also discuss some immediate implications of this result for queues fed by scheduled traffic, including a heavy-traffic limit theorem.

Uniqueness of quasistationary distributions and discrete spectra when ∞ is an entrance boundary and 0 is singular

Thu, 06 Sep 2012 14:16 EDT

Jorge Littin C.

Source: J. Appl. Probab., Volume 49, Number 3, 719--730.

Abstract:
We study quasistationary distributions on a drifted Brownian motion killed at 0, when +∞ is an entrance boundary and 0 is an exit boundary. We prove the existence of a unique quasistationary distribution and of the Yaglom limit.

Probabilities of competing binomial random variables

Thu, 06 Sep 2012 14:16 EDT

Wenbo V. Li, Vladislav V. Vysotsky

Source: J. Appl. Probab., Volume 49, Number 3, 731--744.

Abstract:
Suppose that both you and your friend toss an unfair coin n times, for which the probability of heads is equal to α. What is the probability that you obtain at least d more heads than your friend if you make r additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of n , and demonstrate surprising phase transition phenomenon as the parameters d , r , and α vary. Our main tools are integral representations based on Fourier analysis.

A Pólya approximation to the Poisson-binomial law

Thu, 06 Sep 2012 14:16 EDT

Max Skipper

Source: J. Appl. Probab., Volume 49, Number 3, 745--757.

Abstract:
Using Stein's method, we derive explicit upper bounds on the total variation distance between a Poisson-binomial law (the distribution of a sum of independent but not necessarily identically distributed Bernoulli random variables) and a Pólya distribution with the same support, mean, and variance; a nonuniform bound on the pointwise distance between the probability mass functions is also given. A numerical comparison of alternative distributional approximations on a somewhat representative collection of case studies is also exhibited. The evidence proves that no single one is uniformly most accurate, though it suggests that the Pólya approximation might be preferred in several parameter domains encountered in practice.

Joint distributions of counts of strings in finite Bernoulli sequences

Thu, 06 Sep 2012 14:16 EDT

Fred W. Huffer, Jayaram Sethuraman

Source: J. Appl. Probab., Volume 49, Number 3, 758--772.

Abstract:
An infinite sequence ( Y 1 , Y 2 ,...) of independent Bernoulli random variables with P( Y i = 1) = a / ( a + b + i - 1), i = 1, 2,..., where a > 0 and b ≥ 0, will be called a Bern( a , b ) sequence. Consider the counts Z 1 , Z 2 , Z 3 ,... of occurrences of patterns or strings of the form {11}, {101}, {1001},..., respectively, in this sequence. The joint distribution of the counts Z 1 , Z 2 ,... in the infinite Bern( a , b ) sequence has been studied extensively. The counts from the initial finite sequence ( Y 1 , Y 2 ,..., Y n ) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern( a , 0) and the factorial moments of Z 1 , the count of the string {1, 1}, for a general Bern( a , b ). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

The probability of the Alabama paradox

Thu, 06 Sep 2012 14:16 EDT

Svante Janson, Svante Linusson

Source: J. Appl. Probab., Volume 49, Number 3, 773--794.

Abstract:
Hamilton's method is a natural and common method to distribute seats proportionally between states (or parties) in a parliament. In the USA it has been abandoned due to some drawbacks, in particular the possibility of the Alabama paradox, but it is still in use in many other countries. In this paper we give, under certain assumptions, a closed formula for the asymptotic probability, as the number of seats tends to infinity, that the Alabama paradox occurs given the vector p 1 ,..., p m of relative sizes of the states. From the formula we deduce a number of consequences. For example, the expected number of states that will suffer from the Alabama paradox is asymptotically bounded above by 1 / e and on average approximately 0.123.

On ε-optimality of the pursuit learning algorithm

Thu, 06 Sep 2012 14:16 EDT

Ryan Martin, Omkar Tilak

Source: J. Appl. Probab., Volume 49, Number 3, 795--805.

Abstract:
Estimator algorithms in learning automata are useful tools for adaptive, real-time optimization in computer science and engineering applications. In this paper we investigate theoretical convergence properties for a special case of estimator algorithms - the pursuit learning algorithm. We identify and fill a gap in existing proofs of probabilistic convergence for pursuit learning. It is tradition to take the pursuit learning tuning parameter to be fixed in practical applications, but our proof sheds light on the importance of a vanishing sequence of tuning parameters in a theoretical convergence analysis.

Predicting the supremum: optimality of 'stop at once or not at all'

Thu, 06 Sep 2012 14:16 EDT

Pieter C. Allaart

Source: J. Appl. Probab., Volume 49, Number 3, 806--820.

Abstract:
Let ( X t ) 0 ≤ t ≤ T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process ( X t ) 'as close as possible' to its eventual supremum M T := sup 0 ≤ t ≤ T X t , when the reward for stopping at time τ ≤ T is a nonincreasing convex function of M T - X τ . Under fairly general conditions on the process ( X t ), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T . For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.

The noisy secretary problem and some results on extreme concomitant variables

Thu, 06 Sep 2012 14:16 EDT

Abba M. Krieger, Ester Samuel-Cahn

Source: J. Appl. Probab., Volume 49, Number 3, 821--837.

Abstract:
The classical secretary problem for selecting the best item is studied when the actual values of the items are observed with noise. One of the main appeals of the secretary problem is that the optimal strategy is able to find the best observation with a nontrivial probability of about 0.37, even when the number of observations is arbitrarily large. The results are strikingly different when the qualities of the secretaries are observed with noise. If there is no noise then the only information that is needed is whether an observation is the best among those already observed. Since the observations are assumed to be independent and identically distributed, the solution to this problem is distribution free. In the case of noisy data, the results are no longer distribution free. Furthermore, we need to know the rank of the noisy observation among those already observed. Finally, the probability of finding the best secretary often goes to 0 as the number of observations, n , goes to ∞. The results heavily depend on the behavior of p n , the probability that the observation that is best among the noisy observations is also best among the noiseless observations. Results involving optimal strategies if all that is available is noisy data are described and examples are given to elucidate the results.

Option pricing driven by a telegraph process with random jumps

Thu, 06 Sep 2012 14:16 EDT

Oscar López, Nikita Ratanov

Source: J. Appl. Probab., Volume 49, Number 3, 838--849.

Abstract:
In this paper we propose a class of financial market models which are based on telegraph processes with alternating tendencies and jumps. It is assumed that the jumps have random sizes and that they occur when the tendencies are switching. These models are typically incomplete, but the set of equivalent martingale measures can be described in detail. We provide additional suggestions which permit arbitrage-free option prices as well as hedging strategies to be obtained.

Generalized telegraph process with random delays

Thu, 06 Sep 2012 14:16 EDT

Daoud Bshouty, Antonio Di Crescenzo, Barbara Martinucci, Shelemyahu Zacks

Source: J. Appl. Probab., Volume 49, Number 3, 850--865.

Abstract:
The paper studies the distribution of the location, at time t , of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of U , V and W are absolutely continuous. The velocities are v = +1 upwards, v = -1 downwards, and v = 0 during idle periods. Let Y + ( t ), Y - ( t ) and Y 0 ( t ) denote the total time in (0, t ) of movements upwards, downwards and no movements, respectively. The exact distributions of Y + ( t ) is derived. We also obtain the probability law of X ( t ) = Y + ( t ) - Y - ( t ), which describes the particle's location at time t . Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).

A note on 'Improved Fréchet bounds and model-free pricing of multi-asset options' by Tankov (2011)

Thu, 06 Sep 2012 14:16 EDT

Carole Bernard, Xiao Jiang, Steven Vanduffel

Source: J. Appl. Probab., Volume 49, Number 3, 866--875.

Abstract:
Tankov (2011) improved the Fréchet bounds for a bivariate copula when its values on a compact subset of [0, 1] 2 are given. He showed that the best possible bounds are quasi-copulas and gave a sufficient condition for these bounds to be copulas. In this note we give weaker sufficient conditions to ensure that the bounds are copulas. We also show how this can be useful in portfolio selection. It turns out that finding a copula as a lower bound plays a key role in determining optimal investment strategies explicitly for investors with some type of state-dependent constraints.

A new proof of the Wiener-Hopf factorization via Basu's theorem

Thu, 06 Sep 2012 14:16 EDT

Brian Fralix, Colin Gallagher

Source: J. Appl. Probab., Volume 49, Number 3, 876--882.

Abstract:
We illustrate how Basu's theorem can be used to derive the spatial version of the Wiener-Hopf factorization for a specific class of piecewise-deterministic Markov processes. The classical factorization results for both random walks and Lévy processes follow immediately from our result. The approach is particularly elegant when used to establish the factorization for spectrally one-sided Lévy processes.

The class of distributions associated with the generalized Pollaczek-Khinchine formula

Thu, 06 Sep 2012 14:16 EDT

Offer Kella

Source: J. Appl. Probab., Volume 49, Number 3, 883--887.

Abstract:
The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

A branching process for virus survival

Thu, 06 Sep 2012 14:16 EDT

J. Theodore Cox, Rinaldo B. Schinazi

Source: J. Appl. Probab., Volume 49, Number 3, 888--894.

Abstract:
Quasispecies theory predicts that there is a critical mutation probability above which a viral population will go extinct. Above this threshold the virus loses the ability to replicate the best-adapted genotype, leading to a population composed of low replicating mutants that is eventually doomed. We propose a new branching model that shows that this is not necessarily so. That is, a population composed of ever changing mutants may survive.

Distribution of minimal path lengths when edge lengths are independent heterogeneous exponential random variables

Thu, 06 Sep 2012 14:16 EDT

Sheldon M. Ross

Source: J. Appl. Probab., Volume 49, Number 3, 895--900.

Abstract:
We find the joint distribution of the lengths of the shortest paths from a specified node to all other nodes in a network in which the edge lengths are assumed to be independent heterogeneous exponential random variables. We also give an efficient way to simulate these lengths that requires only one generated exponential per node, as well as efficient procedures to use the simulated data to estimate quantities of the joint distribution.

Functional relationships between price and volatility jumps and their consequences for discretely observed data

Wed, 05 Dec 2012 09:11 EST

Jean Jacod, Claudia Klüppelberg, Gernot Müller

Source: J. Appl. Probab., Volume 49, Number 4, 901--914.

Abstract:
Many prominent continuous-time stochastic volatility models exhibit certain functional relationships between price jumps and volatility jumps. We show that stochastic volatility models like the Ornstein--Uhlenbeck and other continuous-time CARMA models as well as continuous-time GARCH and EGARCH models all exhibit such functional relations. We investigate the asymptotic behaviour of certain functionals of price and volatility processes for discrete observations of the price process on a grid, which are relevant for estimation and testing problems.

The time to ruin in some additive risk models with random premium rates

Wed, 05 Dec 2012 09:11 EST

Martin Jacobsen

Source: J. Appl. Probab., Volume 49, Number 4, 915--938.

Abstract:
The risk processes considered in this paper are generated by an underlying Markov process with a regenerative structure and an independent sequence of independent and identically distributed claims. Between the arrivals of claims the process increases at a rate which is a nonnegative function of the present value of the Markov process. The intensity for a claim to occur is another nonnegative function of the value of the Markov process. The claim arrival times are the regeneration times for the Markov process. Two-sided claims are allowed, but the distribution of the positive claims is assumed to have a Laplace transform that is a rational function. The main results describe the joint Laplace transform of the time at ruin and the deficit at ruin. The method used consists in finding partial eigenfunctions for the generator of the joint process consisting of the Markov process and the accumulated claims process, a joint process which is also Markov. These partial eigenfunctions are then used to find a martingale that directly leads to an expression for the desired Laplace transform. In the final section, three examples are given involving different types of the underlying Markov process.

Asymptotic ruin probabilities for a bivariate Lévy-driven risk model with heavy-tailed claims and risky investments

Wed, 05 Dec 2012 09:11 EST

Xuemiao Hao, Qihe Tang

Source: J. Appl. Probab., Volume 49, Number 4, 939--953.

Abstract:
Consider a general bivariate Lévy-driven risk model. The surplus process Y , starting with Y 0 = x > 0, evolves according to d Y t = Y t - d R t -d P t for t > 0, where P and R are two independent Lévy processes respectively representing a loss process in a world without economic factors and a process describing the return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x →∞, which confirms Paulsen's conjecture.

Ruin probabilities in a finite-horizon risk model with investment and reinsurance

Wed, 05 Dec 2012 09:11 EST

R. Romera, W. Runggaldier

Source: J. Appl. Probab., Volume 49, Number 4, 954--966.

Abstract:
A finite-horizon insurance model is studied where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Our setting is innovative in the sense that we describe in a unified way the timing of the events, that is, the arrivals of claims and the changes of the prices in the financial market, by means of a continuous-time semi-Markov process which appears to be more realistic than, say, classical diffusion-based models. Obtaining explicit optimal solutions for the minimizing ruin probability is a difficult task. Therefore we derive a specific methodology, based on recursive relations for the ruin probability, to obtain a reinsurance and investment policy that minimizes an exponential bound (Lundberg-type bound) on the ruin probability.

Optimal design of dynamic default risk measures

Wed, 05 Dec 2012 09:11 EST

Leo Shen, Robert Elliott

Source: J. Appl. Probab., Volume 49, Number 4, 967--977.

Abstract:
We consider the question of an optimal transaction between two investors to minimize their risks. We define a dynamic entropic risk measure using backward stochastic differential equations related to a continuous-time single jump process. The inf-convolution of dynamic entropic risk measures is a key transformation in solving the optimization problem.

Berry--Esseen bounds and the law of the iterated logarithm for estimators of parameters in an Ornstein--Uhlenbeck process with linear drift

Wed, 05 Dec 2012 09:11 EST

Hui Jiang

Source: J. Appl. Probab., Volume 49, Number 4, 978--989.

Abstract:
We study the asymptotic behaviors of estimators of the parameters in an Ornstein--Uhlenbeck process with linear drift, such as the law of the iterated logarithm (LIL) and Berry--Esseen bounds. As an application of the Berry--Esseen bounds, the precise rates in the LIL for the estimators are obtained.

On the exponential ergodicity of Lévy-driven Ornstein--Uhlenbeck processes

Wed, 05 Dec 2012 09:11 EST

Jian Wang

Source: J. Appl. Probab., Volume 49, Number 4, 990--1004.

Abstract:
Based on the explicit coupling property, the ergodicity and the exponential ergodicity of Lévy-driven Ornstein--Uhlenbeck processes are established.

Spectrally negative Lévy processes perturbed by functionals of their running supremum

Wed, 05 Dec 2012 09:11 EST

Andreas E. Kyprianou, Curdin Ott

Source: J. Appl. Probab., Volume 49, Number 4, 1005--1014.

Abstract:
In the setting of the classical Cramér--Lundberg risk insurance model, Albrecher and Hipp (2007) introduced the idea of tax payments. More precisely, if X = { X t : t #x2264; 0} represents the Cramér--Lundberg process and, for all t #x2264; 0, S t =sup_{ s ≥ t } X s , then Albrecher and Hipp studied X t - γ S t , t #x2264; 0, where γ∈(0,1) is the rate at which tax is paid. This model has been generalised to the setting that X is a spectrally negative Lévy process by Albrecher, Renaud and Zhou (2008). Finally, Kyprianou and Zhou (2009) extended this model further by allowing the rate at which tax is paid with respect to the process S = { S t : t #x2264; 0} to vary as a function of the current value of S . Specifically, they considered the so-called perturbed spectrally negative Lévy process, U t := X t -∫ (0, t ] γ( S _u)d S u , t #x2264; 0, under the assumptions that γ:[0,∞)→ [0,1) and ∫ 0 ∞ (1-γ(s))d s =∞. In this article we show that a number of the identities in Kyprianou and Zhou (2009) are still valid for a much more general class of rate functions γ:[0,∞)→∝. Moreover, we show that, with appropriately chosen γ, the perturbed process can pass continuously (i.e. creep) into (-∞, 0) in two different ways.

On a new class of tempered stable distributions: moments and regular variation

Wed, 05 Dec 2012 09:11 EST

Michael Grabchak

Source: J. Appl. Probab., Volume 49, Number 4, 1015--1035.

Abstract:
We extend the class of tempered stable distributions, which were first introduced in Rosiński (2007). Our new class allows for more structure and more variety of the tail behaviors. We discuss various subclasses and the relations between them. To characterize the possible tails, we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs.

Approximating quasistationary distributions of birth--death processes

Wed, 05 Dec 2012 09:11 EST

Damian Clancy

Source: J. Appl. Probab., Volume 49, Number 4, 1036--1051.

Abstract:
For a sequence of finite state space birth--death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth--death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl--Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.

On the value function of the M/G/1 FCFS and LCFS queues

Wed, 05 Dec 2012 09:11 EST

Esa Hyytiä, Samuli Aalto, Aleksi Penttinen, Jorma Virtamo

Source: J. Appl. Probab., Volume 49, Number 4, 1052--1071.

Abstract:
We consider a single-server queue with Poisson input operating under first-come--first-served (FCFS) or last-come--first-served (LCFS) disciplines. The service times of the customers are independent and obey a general distribution. The system is subject to costs for holding a customer per unit of time, which can be customer specific or customer class specific. We give general expressions for the corresponding value functions, which have elementary compact forms, similar to the Pollaczek--Khinchine mean value formulae. The results generalize earlier work where similar expressions have been obtained for specific service time distributions. The obtained value functions can be readily applied to develop nearly optimal dispatching policies for a broad range of systems with parallel queues, including multiclass scenarios and the cases where service time estimates are available.

Discounted continuous-time controlled Markov chains: convergence of control models

Wed, 05 Dec 2012 09:11 EST

Tomás Prieto-Rumeau, Onésimo Hernández-Lerma

Source: J. Appl. Probab., Volume 49, Number 4, 1072--1090.

Abstract:
We are interested in continuous-time, denumerable state controlled Markov chains (CMCs), with compact Borel action sets, and possibly unbounded transition and reward rates, under the discounted reward optimality criterion. For such CMCs, we propose a definition of a sequence of control models {ℳ n } converging to a given control model ℳ, which ensures that the discount optimal reward and policies of ℳ n converge to those of ℳ. As an application, we propose a finite-state and finite-action truncation technique of the original control model ℳ, which is illustrated by approximating numerically the optimal reward and policy of a controlled population system with catastrophes. We study the corresponding convergence rates.

On the functional central limit theorem for reversible Markov chains with nonlinear growth of the variance

Wed, 05 Dec 2012 09:11 EST

Martial Longla, Costel Peligrad, Magda Peligrad

Source: J. Appl. Probab., Volume 49, Number 4, 1091--1105.

Abstract:
In this paper we study the functional central limit theorem (CLT) for stationary Markov chains with a self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n , and establish that conditional convergence in distribution of partial sums implies the functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

Asymptotics of maxima of strongly dependent Gaussian processes

Wed, 05 Dec 2012 09:11 EST

Zhongquan Tan, Enkelejd Hashorva, Zuoxiang Peng

Source: J. Appl. Probab., Volume 49, Number 4, 1106--1118.

Abstract:
Let { X n ( t ), t ∈[0,∞)}, n ∈ℕ, be standard stationary Gaussian processes. The limit distribution of \sup t ∈[0, T ( n )] | X n ( t )| is established as r n ( t ), the correlation function of { X n ( t ), t ∈[0,∞)}, n ∈ℕ, which satisfies the local and long-range strong dependence conditions, extending the results obtained in Seleznjev (1991).

First passage times of constant-elasticity-of-variance processes with two-sided reflecting barriers

Wed, 05 Dec 2012 09:11 EST

Lijun Bo, Chen Hao

Source: J. Appl. Probab., Volume 49, Number 4, 1119--1133.

Abstract:
In this paper we explore the first passage times of constant-elasticity-of-variance (CEV) processes with two-sided reflecting barriers. The explicit Laplace transforms of the first passage times are derived. Our results can include analytic formulae concerning Laplace transforms of first passage times of reflected Ornstein--Uhlenbeck processes, reflected geometric Brownian motions, and reflected square-root processes.

Upper deviations for split times of branching processes

Wed, 05 Dec 2012 09:11 EST

Hamed Amini, Marc Lelarge

Source: J. Appl. Probab., Volume 49, Number 4, 1134--1143.

Abstract:
Upper deviation results are obtained for the split time of a supercritical continuous-time Markov branching process. More precisely, we establish the existence of logarithmic limits for the likelihood that the split times of the process are greater than an identified value and determine an expression for the limiting quantity. We also give an estimation for the lower deviation probability of the split times, which shows that the scaling is completely different from the upper deviations.

A Glaser twist: focus on the mixture parameters

Wed, 05 Dec 2012 09:11 EST

Henry W. Block, Naftali A. Langberg, Thomas H. Savits

Source: J. Appl. Probab., Volume 49, Number 4, 1144--1155.

Abstract:
In this paper we introduce a variation on Glaser's method for determining the shape of the failure rate function of a mixture. It has often been seen that the shape of the failure rate depends on the mixing parameter q . Our method provides an explanation for this phenomenon. We then illustrate our technique with the mixture of an exponential and a gamma density for all possible cases.

Maximizing the size of the giant

Wed, 05 Dec 2012 09:11 EST

Tom Britton, Pieter Trapman

Source: J. Appl. Probab., Volume 49, Number 4, 1156--1165.

Abstract:
Consider a random graph where the mean degree is given and fixed. In this paper we derive the maximal size of the largest connected component in the graph. We also study the related question of the largest possible outbreak size of an epidemic occurring `on' the random graph (the graph describing the social structure in the community). More precisely, we look at two different classes of random graphs. First, the Poissonian random graph in which each node i is given an independent and identically distributed (i.i.d.) random weight X i with E(X i )=µ, and where there is an edge between i and j with probability 1-e -X i X j /(µ n) , independently of other edges. The second model is the thinned configuration model in which the n vertices of the ground graph have i.i.d. ground degrees, distributed as D , with E( D ) = µ. The graph of interest is obtained by deleting edges independently with probability 1- p . In both models the fraction of vertices in the largest connected component converges in probability to a constant 1- q , where q depends on X or D and p . We investigate for which distributions X and D with given µ and p , 1- q is maximized. We show that in the class of Poissonian random graphs, X should have all its mass at 0 and one other real, which can be explicitly determined. For the thinned configuration model, D should have all its mass at 0 and two subsequent positive integers.

Kronecker-based infinite level-dependent QBD processes

Wed, 05 Dec 2012 09:11 EST

TuǦrul Dayar, M. Can Orhan

Source: J. Appl. Probab., Volume 49, Number 4, 1166--1187.

Abstract:
Markovian systems with multiple interacting subsystems under the influence of a control unit are considered. The state spaces of the subsystems are countably infinite, whereas that of the control unit is finite. A recent infinite level-dependent quasi-birth-and-death model for such systems is extended by facilitating the automatic representation and generation of the nonzero blocks in its underlying infinitesimal generator matrix with sums of Kronecker products. Experiments are performed on systems of stochastic chemical kinetics having two or more countably infinite state space subsystems. Results indicate that, even though more memory is consumed, there are many cases where a matrix-analytic solution coupled with Lyapunov theory yields a faster and more accurate steady-state measure compared to that obtained with simulation.

Improving the Asmusse}--Kroese-type simulation estimators

Wed, 05 Dec 2012 09:11 EST

Samim Ghamami, Sheldon M. Ross

Source: J. Appl. Probab., Volume 49, Number 4, 1188--1193.

Abstract:
The Asmussen--Kroese Monte Carlo estimators of P( S n > u ) and P( S N > u ) are known to work well in rare event settings, where S N is the sum of independent, identically distributed heavy-tailed random variables X 1 ,..., X N and N is a nonnegative, integer-valued random variable independent of the X i . In this paper we show how to improve the Asmussen--Kroese estimators of both probabilities when the X i are nonnegative. We also apply our ideas to estimate the quantity E[( S N - u ) + ].

A note on M/G/1 vacation systems with sojourn time limits

Wed, 05 Dec 2012 09:11 EST

Tsuyoshi Katayama

Source: J. Appl. Probab., Volume 49, Number 4, 1194--1199.

Abstract:
In this paper we deal with an M/G/1 vacation system with the sojourn time (wait plus service) limit and two typical vacation rules, i.e. multiple and single vacation rules. Using the level crossing approach, we derive recursive equations for the steady-state distributions of the virtual waiting times in M/G/1 vacation systems with a general vacation time and two vacation rules.

Optimal scaling of the random walk Metropolis: general criteria for the 0.234 acceptance rule

Wed, 20 Mar 2013 09:01 EDT

Chris Sherlock

Source: J. Appl. Probab., Volume 50, Number 1, 1--15.

Abstract:
Scaling of proposals for Metropolis algorithms is an important practical problem in Markov chain Monte Carlo implementation. Analyses of the random walk Metropolis for high-dimensional targets with specific functional forms have shown that in many cases the optimal scaling is achieved when the acceptance rate is approximately 0.234, but that there are exceptions. We present a general set of sufficient conditions which are invariant to orthonormal transformation of the coordinate axes and which ensure that the limiting optimal acceptance rate is 0.234. The criteria are shown to hold for the joint distribution of successive elements of a stationary p th-order multivariate Markov process.

On the generalized drift Skorokhod problem in one dimension

Wed, 20 Mar 2013 09:01 EDT

Josh Reed, Amy Ward, Dongyuan Zhan

Source: J. Appl. Probab., Volume 50, Number 1, 16--28.

Abstract:
We show how to write the solution to the generalized drift Skorokhod problem in one-dimension in terms of the supremum of the solution of a tractable unrestricted integral equation (that is, an integral equation with no boundaries). As an application of our result, we equate the transient distribution of a reflected Ornstein–Uhlenbeck (OU) process to the first hitting time distribution of an OU process (that is not reflected). Then, we use this relationship to approximate the transient distribution of the GI/GI/1 + GI queue in conventional heavy traffic and the M/M/ N / N queue in a many-server heavy traffic regime.

Optimal sequential change detection for fractional diffusion-type processes

Wed, 20 Mar 2013 09:01 EDT

Alexandra Chronopoulou, Georgios Fellouris

Source: J. Appl. Probab., Volume 50, Number 1, 29--41.

Abstract:
The problem of detecting an abrupt change in the distribution of an arbitrary, sequentially observed, continuous-path stochastic process is considered and the optimality of the CUSUM test is established with respect to a modified version of Lorden's criterion. We apply this result to the case that a random drift emerges in a fractional Brownian motion and we show that the CUSUM test optimizes Lorden's original criterion when a fractional Brownian motion with Hurst index H adopts a polynomial drift term with exponent H +1/2.

Sharp bounds for sums of dependent risks

Wed, 20 Mar 2013 09:01 EDT

Giovanni Puccetti, Ludger Rüschendorf

Source: J. Appl. Probab., Volume 50, Number 1, 42--53.

Abstract:
Sharp tail bounds for the sum of d random variables with given marginal distributions and arbitrary dependence structure have been known since Makarov (1981) and Rüschendorf (1982) for d =2 and, in some examples, for d ≥3. Based on a duality result, dual bounds have been introduced in Embrechts and Puccetti (2006b). In the homogeneous case,\break $ F 1 =···= F n , with monotone density, sharp tail bounds were recently found in Wang and Wang (2011). In this paper we establish the sharpness of the dual bounds in the homogeneous case under general conditions which include, in particular, the case of monotone densities and concave densities. We derive the corresponding optimal couplings and also give an effective method to calculate the sharp bounds.

Limit theorems for a generalized Feller game

Wed, 20 Mar 2013 09:01 EDT

Keisuke Matsumoto, Toshio Nakata

Source: J. Appl. Probab., Volume 50, Number 1, 54--63.

Abstract:
In this paper we study limit theorems for the Feller game which is constructed from one-dimensional simple symmetric random walks, and corresponds to the St. Petersburg game . Motivated by a generalization of the St. Petersburg game which was investigated by Gut (2010), we generalize the Feller game by introducing the parameter α. We investigate limit distributions of the generalized Feller game corresponding to the results of Gut. Firstly, we give the weak law of large numbers for α=1. Moreover, for 0<α≤1, we have convergence in distribution to a stable law with index α. Finally, some limit theorems for a polynomial size and a geometric size deviation are given.

Asymptotics for the first passage times of Lévy processes and random walks

Wed, 20 Mar 2013 09:01 EDT

Denis Denisov, Vsevolod Shneer

Source: J. Appl. Probab., Volume 50, Number 1, 64--84.

Abstract:
We study the exact asymptotics for the distribution of the first time, τ x , a Lévy process X t crosses a fixed negative level - x . We prove that ℙ{τ x > t } ~ V ( x ) ℙ{ X t ≥0}/ t as t →∞ for a certain function V ( x ). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τ x > t } explicitly in both light- and heavy-tailed cases.

Random walks reaching against all odds the other side of the quarter plane

Wed, 20 Mar 2013 09:01 EDT

Johan S. H. van Leeuwaarden, Kilian Raschel

Source: J. Appl. Probab., Volume 50, Number 1, 85--102.

Abstract:
For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state ( i 0 , j 0 ), we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive a certain integral representation for the probability of this event, and an asymptotic expression for the case when i 0 becomes large, a situation in which the event becomes highly unlikely. The integral representation follows from the solution of a boundary value problem and involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model, and the asymmetric exclusion process.

Asymptotics of hybrid fluid queues with Lévy input

Wed, 20 Mar 2013 09:01 EDT

Krzysztof Dębicki, Iwona Sierpińska, Bert Zwart

Source: J. Appl. Probab., Volume 50, Number 1, 103--113.

Abstract:
Let { X ( t ): t ∈ℝ} be the integrated on–off process with regularly varying on-periods, and let { Y ( t ): t ∈ℝ} be a centered Lévy process with regularly varying positive jumps (independent of X (·)). We study the exact asymptotics of ℙ(sup t ≥0 { X ( t )+ Y ( t )- ct }> u ) as u →∞, with special attention to the case r=c , where r is the increase rate of the on–off process during the on-periods.

Domain of attraction of the quasistationary distribution for birth-and-death processes

Wed, 20 Mar 2013 09:01 EDT

Hanjun Zhang, Yixia Zhu

Source: J. Appl. Probab., Volume 50, Number 1, 114--126.

Abstract:
We consider a birth–death process { X ( t ), t ≥0} on the positive integers for which the origin is an absorbing state with birth coefficients λ n , n ≥0, and death coefficients μ n , n ≥0. If we define A =∑ n =1 ∞ 1/λ n π n and S =∑ n =1 ∞ (1/λ n π n )∑ i = n +1 ∞ π i , where {π n , n ≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A =∞ and S <∞, then λ C >0 and there is precisely one quasistationary distribution, namely, { a j (λ C )}, where λ C is the decay parameter of { X ( t ), t ≥0} in C ={1,2,...} and a j ( x )≡μ 1 -1 π j xQ j ( x ), j= 1,2,... . In this paper we prove that there is a unique quasistationary distribution that attracts all initial distributions supported in C , if and only if the birth–death process { X ( t ), t ≥0} satisfies both A =∞ and S <∞. That is, for any probability measure M ={ m i , i =1,2,...}, we have lim t \→∞ ℙ M ( X ( t )= j ∣ T > t )= a j (λ C ), j= 1,2,..., where T =inf{ t ≥0 : X ( t )=0} is the extinction time of { X ( t ), t ≥0} if and only if the birth–death process { X ( t ), t ≥0} satisfies both A =∞ and S <∞.

Heavy tails in queueing systems: impact of parallelism on tail performance

Wed, 20 Mar 2013 09:01 EDT

Bo Jiang, Jian Tan, Wei Wei, Ness Shroff, Don Towsley

Source: J. Appl. Probab., Volume 50, Number 1, 127--150.

Abstract:
In this paper we quantify the efficiency of parallelism in systems that are prone to failures and exhibit power law processing delays. We characterize the performance of two prototype schemes of parallelism, redundant and split , in terms of both the power law exponent and exact asymptotics of the delay distribution tail. We also develop the optimal splitting scheme which ensures that split always outperforms redundant.

Computing stationary expectations in level-dependent QBD processes

Wed, 20 Mar 2013 09:01 EDT

Hendrik Baumann, Werner Sandmann

Source: J. Appl. Probab., Volume 50, Number 1, 151--165.

Abstract:
Stationary expectations corresponding to long-run averages of additive functionals on level-dependent quasi-birth-and-death processes are considered. Special cases include long-run average costs or rewards, moments and cumulants of steady-state queueing network performance measures, and many others. We provide a matrix-analytic scheme for numerically computing such stationary expectations without explicitly computing the stationary distribution of the process, which yields an algorithm that is as quick as its counterparts for stationary distributions but requires far less computer storage. Specific problems arising in the case of infinite state spaces are discussed and the application of the algorithm is demonstrated by a queueing network example.

Conditional distributions of processes related to fractional Brownian motion

Wed, 20 Mar 2013 09:01 EDT

Holger Fink, Claudia Klüppelberg, Martina Zähle

Source: J. Appl. Probab., Volume 50, Number 1, 166--183.

Abstract:
Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

Regular perturbation of V -geometrically ergodic Markov chains

Wed, 20 Mar 2013 09:01 EDT

Déborah Ferré, Loïc Hervé, James Ledoux

Source: J. Appl. Probab., Volume 50, Number 1, 184--194.

Abstract:
In this paper, new conditions for the stability of V -geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher regularity properties are investigated. As an illustration, an asymptotic expansion of the invariant probability measure for an autoregressive model with independent and identically distributed noises (with a nonstandard probability density function) is obtained.

A time-homogeneous diffusion model with tax

Wed, 20 Mar 2013 09:01 EDT

Bin Li, Qihe Tang, Xiaowen Zhou

Source: J. Appl. Probab., Volume 50, Number 1, 195--207.

Abstract:
We study the two-sided exit problem of a time-homogeneous diffusion process with tax payments of loss-carry-forward type and obtain explicit formulae for the Laplace transforms associated with the two-sided exit problem. The expected present value of tax payments until default, the two-sided exit probabilities, and, hence, the nondefault probability with the default threshold equal to the lower bound are solved as immediate corollaries. A sufficient and necessary condition for the tax identity in ruin theory is discovered.

Splitting trees stopped when the first clock rings and Vervaat's transformation

Wed, 20 Mar 2013 09:01 EDT

Amaury Lambert, Pieter Trapman

Source: J. Appl. Probab., Volume 50, Number 1, 208--227.

Abstract:
We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let N t denote the population size at time t . We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on { T <∞}, the joint law of ( N t , T , X ( T ) ), where X ( T ) is the jumping contour process of the tree truncated at time T , is equal to that of ( M , -I M , Y' M ) conditional on { M ≠0}. Here M +1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Lévy process Y without negative jumps reflected below its supremum; I M is the infimum of the path Y M , which in turn is defined as Y killed at its last visit of 0 before e ; and Y' M is the Vervaat transform of Y M . This identity yields an explanation for the geometric distribution of N T (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on { N T = n }, and also on { N T = n , T<a }, the ages and residual lifetimes of the n alive individuals at time T are i.i.d. and independent of n . We provide explicit formulae for this distribution and give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital.

Asymptotic analysis of Hoppe trees

Wed, 20 Mar 2013 09:01 EDT

Kevin Leckey, Ralph Neininger

Source: J. Appl. Probab., Volume 50, Number 1, 228--238.

Abstract:
We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node, where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight ϑ>0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For ϑ=1, the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length, and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n →∞. Mainly expectations, variances, and asymptotic distributions of these parameters are derived.

Ancestral graph with bias in gene conversion

Wed, 20 Mar 2013 09:01 EDT

Shuhei Mano

Source: J. Appl. Probab., Volume 50, Number 1, 239--255.

Abstract:
Gene conversion is a genetic mechanism by which one gene is `copied and pasted' onto another gene, where the direction can be biased between the different types. In this paper, a stochastic model for biased gene conversion within a d -unlinked multigene family and its diffusion approximation are developed for a finite Moran population. A connection with a d -island model is made. A formula for the fixation probability in the absence of mutation is given. A two-timescale argument is applied in the case of the strong conversion limit. The dual process is generally shown to be a biased voter model, which generates an ancestral bias graph for a given sample. An importance sampling algorithm for computing the likelihood of the sample is deduced.

Duality between the two-locus Wright–Fisher diffusion model and the ancestral process with recombination

Wed, 20 Mar 2013 09:01 EDT

Shuhei Mano

Source: J. Appl. Probab., Volume 50, Number 1, 256--271.

Abstract:
Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

Dynamic signatures of coherent systems based on sequential order statistics

Wed, 20 Mar 2013 09:01 EDT

M. Burkschat, J. Navarro

Source: J. Appl. Probab., Volume 50, Number 1, --.

Abstract:
Sequential order statistics can be used to describe the ordered lifetimes of components in a system, where the failure of a component may affect the performance of remaining components. In this paper mixture representations of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. Stochastic comparisons of differently structured systems are obtained and properties of the weights in the mixture representations are examined. Furthermore, corresponding representations of the residual lifetime and the inactivity time of a system given the additional information about a previous failure time are derived.

Asymptotic expected number of passages of a random walk through an interval

Wed, 20 Mar 2013 09:01 EDT

Offer Kella, Wolfgang Stadje

Source: J. Appl. Probab., Volume 50, Number 1, 288--294.

Abstract:
In this note we find a new result concerning the asymptotic expected number of passages of a finite or infinite interval ( x,x+h ) as x→∞ for a random walk with increments having a positive expected value. If the increments are distributed like X then the limit for 0< h <∞ turns out to have the form Emin(| X |, h )/E X , which unexpectedly is independent of h for the special case where | X |≤ b <∞ almost surely and h>b . When h =∞, the limit is Emax( X ,0)/E X . For the case of a simple random walk, a more pedestrian derivation of the limit is given.

The Laplace transform of hitting times of integrated geometric Brownian motion

Wed, 20 Mar 2013 09:01 EDT

Adam Metzler

Source: J. Appl. Probab., Volume 50, Number 1, 295--299.

Abstract:
In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

A duality relation between the workload and attained waiting time in FCFS G/G/ s queues

Wed, 20 Mar 2013 09:01 EDT

Yi-Ching Yao

Source: J. Appl. Probab., Volume 50, Number 1, 300--307.

Abstract:
Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given sample path is identical to that of the attained waiting time values over the same period. For a given sample path of an FCFS G/G/ s queue, we construct a dual sample path of a dual queue which is FCFS G/G/ s in reverse time. It is shown that the workload process on the original sample path is identical to the total attained waiting time process on the dual sample path. As an application of this duality relation, we show that, for a time-stationary FCFS M/M/ s / k queue, the workload process is equal in distribution to the time-reversed total attained waiting time process.