Advances in Applied Probability Articles (Project Euclid)

Excursion sets of three classes of stable random fields

Thu, 05 Aug 2010 15:41 EDT

Robert J. Adler, Gennady Samorodnitsky, Jonathan E. Taylor

Source: Adv. in Appl. Probab., Volume 42, Number 2, 293--318.

Abstract:
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.

Limiting distributions for a class of diminishing urn models

Thu, 08 Mar 2012 09:25 EST

Markus Kuba, Alois Panholzer

Source: Adv. in Appl. Probab., Volume 44, Number 1, 87--116.

Abstract:
In this work we analyze a class of 2 x 2 Pólya-Eggenberger urn models with ball replacement matrix M = (- a 0 \\ c - d ), a , d ∈ N and c = pa with p ∈ N 0 . We determine limiting distributions by obtaining a precise recursive description of the moments of the considered random variables, which allows us to deduce asymptotic expansions of the moments. In particular, we obtain limiting distributions for the pills problem a = c = d = 1, originally proposed by Knuth and McCarthy. Furthermore, we also obtain limiting distributions for the well-known sampling without replacement urn, a = d = 1 and c = 0, and generalizations of it to arbitrary a , d ∈ N and c = 0. Moreover, we obtain a recursive description of the moment sequence for a generalized problem.

Pareto Lévy measures and multivariate regular variation

Thu, 08 Mar 2012 09:25 EST

Irmingard Eder, Claudia Klüppelberg

Source: Adv. in Appl. Probab., Volume 44, Number 1, 117--138.

Abstract:
We consider regular variation of a Lévy process X := ( X _t) t ≥0 in R d with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure , which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.

Joint vertex degrees in the inhomogeneous random graph model G ( n , { p ij })

Thu, 08 Mar 2012 09:25 EST

Kaisheng Lin, Gesine Reinert

Source: Adv. in Appl. Probab., Volume 44, Number 1, 139--165.

Abstract:
In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

The coupon collector's problem revisited: asymptotics of the variance

Thu, 08 Mar 2012 09:25 EST

Aristides V. Doumas, Vassilis G. Papanicolaou

Source: Adv. in Appl. Probab., Volume 44, Number 1, 166--195.

Abstract:
We develop techniques for computing the asymptotics of the first and second moments of the number T N of coupons that a collector has to buy in order to find all N existing different coupons as N → ∞. The probabilities (occurring frequencies) of the coupons can be quite arbitrary. From these asymptotics we obtain the leading behavior of the variance V [ T N ] of T N (see Theorems 3.1 and 4.4). Then, we combine our results with the general limit theorems of Neal in order to derive the limit distribution of T N (appropriately normalized), which, for a large class of probabilities, turns out to be the standard Gumbel distribution. We also give various illustrative examples.

Numerical methods for the exit time of a piecewise-deterministic Markov process

Thu, 08 Mar 2012 09:25 EST

Adrien Brandejsky, Benoîte De Saporta, François Dufour

Source: Adv. in Appl. Probab., Volume 44, Number 1, 196--225.

Abstract:
We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. We give an academic example and a model from the reliability field to illustrate the results of the paper.

Extinction probability of interacting branching collision processes

Thu, 08 Mar 2012 09:25 EST

Anyue Chen, Junping Li, Yiqing Chen, Dingxuan Zhou

Source: Adv. in Appl. Probab., Volume 44, Number 1, 226--259.

Abstract:
We consider the uniqueness and extinction properties of the interacting branching collision process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process and a collision branching process. We establish that there is a unique IBCP, and derive necessary and sufficient conditions for it to be nonexplosive that are easily checked. Explicit expressions are obtained for the extinction probabilities for both regular and irregular cases. The associated expected hitting times are also considered. Examples are provided to illustrate our results.

The distribution of Foschini's lower bound for channel capacity

Thu, 08 Mar 2012 09:25 EST

Christopher S. Withers, Saralees Nadarajah

Source: Adv. in Appl. Probab., Volume 44, Number 1, 260--269.

Abstract:
Foschini gave a lower bound for the channel capacity of an N -transmit M -receive antenna system in a Raleigh fading environment with independence at both transmitters and receivers. We show that this bound is approximately normal.

Asymptotic conditional distribution of exceedance counts

Thu, 08 Mar 2012 09:25 EST

Michael Falk, Diana Tichy

Source: Adv. in Appl. Probab., Volume 44, Number 1, 270--291.

Abstract:
We investigate the asymptotic distribution of the number of exceedances among d identically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of the fragility index , which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the first d RVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence as d increases.

Bounds for the availabilities of multistate monotone systems based on decomposition into stochastically independent modules

Thu, 08 Mar 2012 09:25 EST

J. Gåsemyr

Source: Adv. in Appl. Probab., Volume 44, Number 1, 292--308.

Abstract:
Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities and unavailabilities of the system in an interval, i.e. the probabilities that the system performs above or below the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities and unavailabilities exactly, but it is possible to construct lower and upper bounds based on the minimal path and cut vectors to the different levels. In this paper we consider systems which allow a modular decomposition. We analyse in depth the relationship between the minimal path and cut vectors for the system, the modules, and the organizing structure. We analyse the extent to which the availability bounds are improved by taking advantage of the modular decomposition. This problem was also treated in Butler (1982) and Funnemark and Natvig (1985), but the treatment was based on an inadequate analysis of the relationship between the different minimal path and cut vectors involved, and as a result was somewhat inaccurate. We also extend to interval bounds that have previously only been given for availabilities at a fixed point of time.

Correction: Stochastic and deterministic analysis of SIS household epidemics

Thu, 08 Mar 2012 09:25 EST

P. Neal

Source: Adv. in Appl. Probab., Volume 44, Number 1, 309--310.

On statistical properties of sets fulfilling rolling-type conditions

Sat, 16 Jun 2012 16:32 EDT

A. Cuevas, R. Fraiman, B. Pateiro-López

Source: Adv. in Appl. Probab., Volume 44, Number 2, 311--329.

Abstract:
Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r -convexity, and the rolling condition. First, the relations between these shape conditions are analyzed. Second, for the estimation of sets fulfilling a rolling condition, we obtain a result of 'full consistency' (i.e. consistency with respect to the Hausdorff metric for the target set and for its boundary). Third, the class of uniformly bounded compact sets whose reach is not smaller than a given constant r is shown to be a P -uniformity class (in Billingsley and Topsøe's (1967) sense) and, in particular, a Glivenko-Cantelli class. Fourth, under broad conditions, the r -convex hull of the sample is proved to be a fully consistent estimator of an r -convex support in the two-dimensional case. Moreover, its boundary length is shown to converge (almost surely) to that of the underlying support. Fifth, the above results are applied to obtain new consistency statements for level set estimators based on the excess mass methodology (see Polonik (1995)).

Convex hulls of uniform samples from a convex polygon

Sat, 16 Jun 2012 16:32 EDT

Piet Groeneboom

Source: Adv. in Appl. Probab., Volume 44, Number 2, 330--342.

Abstract:
In Groeneboom (1988) a central limit theorem for the number of vertices N n of the convex hull of a uniform sample from the interior of a convex polygon was derived. To be more precise, it was shown that { N n - (2/3) r log n } / {(10/27) r log n } 1/2 converges in law to a standard normal distribution, if r is the number of vertices of the convex polygon from which the sample is taken. In the unpublished preprint Nagaev and Khamdamov (1991) a central limit result for the joint distribution of N n and A n is given, where A n is the area of the convex hull, using a coupling of the sample process near the border of the polygon with a Poisson point process as in Groeneboom (1988), and representing the remaining area in the Poisson approximation as a union of a doubly infinite sequence of independent standard exponential random variables. We derive this representation from the representation in Groeneboom (1988) and also prove the central limit result of Nagaev and Khamdamov (1991), using this representation. The relation between the variances of the asymptotic normal distributions of the number of vertices and the area, established in Nagaev and Khamdamov (1991), corresponds to a relation between the actual sample variances of N n and A n in Buchta (2005). We show how these asymptotic results all follow from one simple guiding principle. This corrects at the same time the scaling constants in Cabo and Groeneboom (1994) and Nagaev (1995).

Stein's method and stochastic orderings

Sat, 16 Jun 2012 16:32 EDT

Fraser Daly, Claude Lefèvre, Sergey Utev

Source: Adv. in Appl. Probab., Volume 44, Number 2, 343--372.

Abstract:
A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s -convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k -runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

Increasing hazard rate of mixtures for natural exponential families

Sat, 16 Jun 2012 16:32 EDT

Shaul K. Bar-Lev, Gérard Letac

Source: Adv. in Appl. Probab., Volume 44, Number 2, 373--390.

Abstract:
Hazard rates play an important role in various areas, e.g. reliability theory, survival analysis, biostatistics, queueing theory, and actuarial studies. Mixtures of distributions are also of great preeminence in such areas as most populations of components are indeed heterogeneous. In this study we present a sufficient condition for mixtures of two elements of the same natural exponential family (NEF) to have an increasing hazard rate. We then apply this condition to some classical NEFs having either quadratic or cubic variance functions (VFs) and others as well. Particular attention is paid to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF, and the NEF generated by Kummer distributions of type 2. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEFs. Various lemmas and propositions are needed to verify this condition for such NEFs. It should be pointed out, however, that our results are mainly applied to a mixture of two populations.

Closed-form asymptotic sampling distributions under the coalescent with recombination for an arbitrary number of loci

Sat, 16 Jun 2012 16:32 EDT

Anand Bhaskar, Yun S. Song

Source: Adv. in Appl. Probab., Volume 44, Number 2, 391--407.

Abstract:
Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on an asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

Approximate sampling formulae for general finite-alleles models of mutation

Sat, 16 Jun 2012 16:32 EDT

Anand Bhaskar, John A. Kamm, Yun S. Song

Source: Adv. in Appl. Probab., Volume 44, Number 2, 408--428.

Abstract:
Many applications in genetic analyses utilize sampling distributions, which describe the probability of observing a sample of DNA sequences randomly drawn from a population. In the one-locus case with special models of mutation, such as the infinite-alleles model or the finite-alleles parent-independent mutation model, closed-form sampling distributions under the coalescent have been known for many decades. However, no exact formula is currently known for more general models of mutation that are of biological interest. In this paper, models with finitely-many alleles are considered, and an urn construction related to the coalescent is used to derive approximate closed-form sampling formulae for an arbitrary irreducible recurrent mutation model or for a reversible recurrent mutation model, depending on whether the number of distinct observed allele types is at most three or four, respectively. It is demonstrated empirically that the formulae derived here are highly accurate when the per-base mutation rate is low, which holds for many biological organisms.

On the joint behavior of types of coupons in generalized coupon collection

Sat, 16 Jun 2012 16:32 EDT

Hosam M. Mahmoud, Robert T. Smythe

Source: Adv. in Appl. Probab., Volume 44, Number 2, 429--451.

Abstract:
The 'coupon collection problem' refers to a class of occupancy problems in which j identical items are distributed, independently and at random, to n cells, with no restrictions on multiple occupancy. Identifying the cells as coupons, a coupon is 'collected' if the cell is occupied by one or more of the distributed items; thus, some coupons may never be collected, whereas others may be collected once or twice or more. We call the number of coupons collected exactly r times coupons of type r . The coupon collection model we consider is general, in that a random number of purchases occurs at each stage of collecting a large number of coupons; the sample sizes at each stage are independent and identically distributed according to a sampling distribution . The joint behavior of the various types is an intricate problem. In fact, there is a variety of joint central limit theorems (and other limit laws) that arise according to the interrelation between the mean, variance, and range of the sampling distribution, and of course the phase (how far we are in the collection processes). According to an appropriate combination of the mean of the sampling distribution and the number of available coupons, the phase is sublinear, linear, or superlinear. In the sublinear phase, the normalization that produces a Gaussian limit law for uncollected coupons can be used to obtain a multivariate central limit law for at most two other types - depending on the rates of growth of the mean and variance of the sampling distribution, we may have a joint central limit theorem between types 0 and 1, or between types 0, 1, and 2. In the linear phase we have a multivariate central limit theorem among the types 0, 1,..., k for any fixed k .

A self-normalized central limit theorem for Markov random walks

Sat, 16 Jun 2012 16:32 EDT

Cheng-Der Fuh, Tian-Xiao Pang

Source: Adv. in Appl. Probab., Volume 44, Number 2, 452--478.

Abstract:
Motivated by the study of the asymptotic normality of the least-squares estimator in the (autoregressive) AR(1) model under possibly infinite variance, in this paper we investigate a self-normalized central limit theorem for Markov random walks. That is, let { X n , n ≥ 0} be a Markov chain on a general state space X with transition probability P and invariant measure π. Suppose that an additive component S n takes values on the real line R , and is adjoined to the chain such that { S n , n ≥ 1} is a Markov random walk. Assume that S n = ∑ k =1 n ξ k , and that {ξ n , n ≥ 1} is a nondegenerate and stationary sequence under π that belongs to the domain of attraction of the normal law with zero mean and possibly infinite variance. By making use of an asymptotic variance formula of S n / √ n , we prove a self-normalized central limit theorem for S n under some regularity conditions. An essential idea in our proof is to bound the covariance of the Markov random walk via a sequence of weight functions, which plays a crucial role in determining the moment condition and dependence structure of the Markov random walk. As illustrations, we apply our results to the finite-state Markov chain, the AR(1) model, and the linear state space model.

Fractional relaxation equations and Brownian crossing probabilities of a random boundary

Sat, 16 Jun 2012 16:32 EDT

L. Beghin

Source: Adv. in Appl. Probab., Volume 44, Number 2, 479--505.

Abstract:
In this paper we analyze different forms of fractional relaxation equations of order ν ∈ (0, 1), and we derive their solutions in both analytical and probabilistic forms. In particular, we show that these solutions can be expressed as random boundary crossing probabilities of various types of stochastic process, which are all related to the Brownian motion $B$. In the special case ν = ½, the fractional relaxation is shown to coincide with Pr{sup 0≤ s ≤ t B ( s ) < U } for an exponential boundary U . When we generalize the distributions of the random boundary, passing from the exponential to the gamma density, we obtain more and more complicated fractional equations.

Asymptotic dependence for light-tailed homothetic densities

Sat, 16 Jun 2012 16:32 EDT

Guus Balkema, Natalia Nolde

Source: Adv. in Appl. Probab., Volume 44, Number 2, 506--527.

Abstract:
Dependence between coordinate extremes is a key factor in any multivariate risk assessment. Hence, it is of interest to know whether the components of a given multivariate random vector exhibit asymptotic independence or asymptotic dependence. In the latter case the structure of the asymptotic dependence has to be clarified. In the multivariate setting it is common to have an explicit form of the density rather than the distribution function. In this paper we therefore give criteria for asymptotic dependence in terms of the density. We consider distributions with light tails and restrict attention to continuous unimodal densities defined on the whole space or on an open convex cone. For simplicity, the density is assumed to be homothetic : all level sets have the same shape. Balkema and Nolde (2010) contains conditions on the shape which guarantee asymptotic independence. The situation for asymptotic dependence, treated in the present paper, is more delicate.

Implicit renewal theory and power tails on trees

Sat, 16 Jun 2012 16:32 EDT

Predrag R. Jelenković, Mariana Olvera-Cravioto

Source: Adv. in Appl. Probab., Volume 44, Number 2, 528--561.

Abstract:
We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to R = D ∑ i =1 N C i R i + Q , R = D (∨ i =1 N C i R i ) ∨ Q , and similar recursions, where ( Q , N , C 1 , C 2 ,...) is a nonnegative random vector with N ∈ {0, 1, 2, 3,...} ∪ {∞}, and { R i } i ∈ N } are independent and identically distributed copies of R , independent of ( Q , N , C 1 , C 2 ,...); here '∨' denotes the maximum operator.

On the transient behavior of Ehrenfest and Engset processes

Sat, 16 Jun 2012 16:32 EDT

Mathieu Feuillet, Philippe Robert

Source: Adv. in Appl. Probab., Volume 44, Number 2, 562--582.

Abstract:
Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. In this paper we investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. We rely on martingale methods; a key ingredient is an important family of simple nonnegative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.

Typical distances in ultrasmall random networks

Sat, 16 Jun 2012 16:32 EDT

Steffen Dereich, Christian Mönch, Peter Mörters

Source: Adv. in Appl. Probab., Volume 44, Number 2, 583--601.

Abstract:
We show that in preferential attachment models with power-law exponent τ ∈ (2, 3) the distance between randomly chosen vertices in the giant component is asymptotically equal to (4 + o (1))log log N / (-log(τ - 2)), where N denotes the number of nodes. This is twice the value obtained for the configuration model with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

Random marked sets

Thu, 06 Sep 2012 14:15 EDT

F. Ballani, Z. Kabluchko, M. Schlather

Source: Adv. in Appl. Probab., Volume 44, Number 3, 603--616.

Abstract:
We aim to link random fields and marked point processes, and, therefore, introduce a new class of stochastic processes which are defined on a random set in R d . Unlike for random fields, the mark covariance function of a random marked set is in general not positive definite. This implies that in many situations the use of simple geostatistical methods appears to be questionable. Surprisingly, for a special class of processes based on Gaussian random fields, we do have positive definiteness for the corresponding mark covariance function and mark correlation function.

Sharpness in the k -nearest-neighbours random geometric graph model

Thu, 06 Sep 2012 14:15 EDT

Victor Falgas-Ravry, Mark Walters

Source: Adv. in Appl. Probab., Volume 44, Number 3, 617--634.

Abstract:
Let S n , k denote the random graph obtained by placing points in a square box of area n according to a Poisson process of intensity 1 and joining each point to its k nearest neighbours. Balister, Bollobás, Sarkar and Walters (2005) conjectured that, for every 0 < ε < 1 and all sufficiently large n , there exists C = C (ε) such that, whenever the probability that S n , k is connected is at least ε, then the probability that S n , k + C is connected is at least 1 - ε. In this paper we prove this conjecture. As a corollary, we prove that there exists a constant C ' such that, whenever k ( n ) is a sequence of integers such that the probability S n , k ( n ) is connected tends to 1 as n → ∞, then, for any integer sequence s ( n ) with s ( n ) = o (log n ), the probability S n , k ( n )+⌊ C ' s log log n ⌋ is s -connected (i.e. remains connected after the deletion of any s - 1 vertices) tends to 1 as n → ∞. This proves another conjecture given in Balister, Bollobás, Sarkar and Walters (2009).

Spatial STIT tessellations: distributional results for I-segments

Thu, 06 Sep 2012 14:15 EDT

Christoph Thäle, Viola Weiss, Werner Nagel

Source: Adv. in Appl. Probab., Volume 44, Number 3, 635--654.

Abstract:
In this paper we consider three-dimensional random tessellations that are stable under iteration (STIT tessellations). STIT tessellations arise as a result of subsequent cell division, which implies that their cells are not face-to-face. The edges of the cell-dividing polygons are the so-called I-segments of the tessellation. The main result is an explicit formula for the distribution of the number of vertices in the relative interior of the typical I-segment. In preparation for its proof, we obtain other distributional identities for the typical I-segment and the length-weighted typical I-segment, which provide new insight into the spatiotemporal construction process.

Optimal stopping problems for asset management

Thu, 06 Sep 2012 14:15 EDT

Savas Dayanik, Masahiko Egami

Source: Adv. in Appl. Probab., Volume 44, Number 3, 655--677.

Abstract:
An asset manager invests the savings of some investors in a portfolio of defaultable bonds. The manager pays the investors coupons at a constant rate and receives a management fee proportional to the value of the portfolio. He/she also has the right to walk out of the contract at any time with the net terminal value of the portfolio after payment of the investors' initial funds, and is not responsible for any deficit. To control the principal losses, investors may buy from the manager a limited protection which terminates the agreement as soon as the value of the portfolio drops below a predetermined threshold. We assume that the value of the portfolio is a jump diffusion process and find an optimal termination rule of the manager with and without protection. We also derive the indifference price of a limited protection. We illustrate the solution method on a numerical example. The motivation comes from the collateralized debt obligations.

Nonlinear filtering for jump diffusion observations

Thu, 06 Sep 2012 14:15 EDT

Claudia Ceci, Katia Colaneri

Source: Adv. in Appl. Probab., Volume 44, Number 3, 678--701.

Abstract:
We deal with the filtering problem of a general jump diffusion process, X , when the observation process, Y , is a correlated jump diffusion process having common jump times with X . In this setting, at any time t the σ-algebra F Y t provides all the available information about X t , and the central goal is to characterize the filter, π t , which is the conditional distribution of X t given observations F Y t . To this end, we prove that π t solves the Kushner-Stratonovich equation and, by applying the filtered martingale problem approach (see Kurtz and Ocone (1988)), that it is the unique weak solution to this equation. Under an additional hypothesis, we also provide a pathwise uniqueness result.

The class of tenable zero-balanced Pólya urn schemes: characterization and Gaussian phases

Thu, 06 Sep 2012 14:15 EDT

Sanaa Kholfi, Hosam M. Mahmoud

Source: Adv. in Appl. Probab., Volume 44, Number 3, 702--728.

Abstract:
We study a class of tenable, irreducible, nondegenerate zero-balanced Pólya urn schemes. We give a full characterization of the class by sufficient and necessary conditions. Only forms with a certain cyclic structure in their replacement matrix are admissible. The scheme has a steady state into proportions governed by the principal (left) eigenvector of the average replacement matrix. We study the gradual change for any such urn containing n → ∞ balls from the initial condition to the steady state. We look at the status of an urn starting with an asymptotically positive proportion of each color after j n draws. We identify three phases of j n : the growing sublinear, the linear, and the superlinear. In the growing sublinear phase the number of balls of different colors has an asymptotic joint multivariate normal distribution, with mean and covariance structure that are influenced by the initial conditions. In the linear phase a different multivariate normal distribution kicks in, in which the influence of the initial conditions is attenuated. The steady state is not a good approximation until a certain superlinear amount of time has elapsed. We give interpretations for how the results in different phases conjoin at the `seam lines'. In fact, these Gaussian phases are all manifestations of one master theorem. The results are obtained via multivariate martingale theory. We conclude with some illustrating examples.

Piecewise-deterministic Markov processes as limits of Markov jump processes

Thu, 06 Sep 2012 14:15 EDT

Uwe Franz, Volkmar Liebscher, Stefan Zeiser

Source: Adv. in Appl. Probab., Volume 44, Number 3, 729--748.

Abstract:
A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology where entities often occur in different scales of numbers.

Averaging for a fully coupled piecewise-deterministic Markov process in infinite dimensions

Thu, 06 Sep 2012 14:15 EDT

Alexandre Genadot, Michèle Thieullen

Source: Adv. in Appl. Probab., Volume 44, Number 3, 749--773.

Abstract:
In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this `two-time-scale' model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

The expected total cost criterion for Markov decision processes under constraints: a convex analytic approach

Thu, 06 Sep 2012 14:15 EDT

Fran\c cois Dufour, M. Horiguchi, A. B. Piunovskiy

Source: Adv. in Appl. Probab., Volume 44, Number 3, 774--793.

Abstract:
This paper deals with discrete-time Markov decision processes (MDPs) under constraints where all the objectives have the same form of expected total cost over the infinite time horizon. The existence of an optimal control policy is discussed by using the convex analytic approach. We work under the assumptions that the state and action spaces are general Borel spaces, and that the model is nonnegative, semicontinuous, and there exists an admissible solution with finite cost for the associated linear program. It is worth noting that, in contrast to the classical results in the literature, our hypotheses do not require the MDP to be transient or absorbing. Our first result ensures the existence of an optimal solution to the linear program given by an occupation measure of the process generated by a randomized stationary policy. Moreover, it is shown that this randomized stationary policy provides an optimal solution to this Markov control problem. As a consequence, these results imply that the set of randomized stationary policies is a sufficient set for this optimal control problem. Finally, our last main result states that all optimal solutions of the linear program coincide on a special set with an optimal occupation measure generated by a randomized stationary policy. Several examples are presented to illustrate some theoretical issues and the possible applications of the results developed in the paper.

Tail behavior of randomly weighted sums

Thu, 06 Sep 2012 14:15 EDT

Rajat Subhra Hazra, Krishanu Maulik

Source: Adv. in Appl. Probab., Volume 44, Number 3, 794--814.

Abstract:
Let { X t , t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θ t , t ≥ 1} be a sequence of positive random variables independent of the sequence { X t , t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X (∞) = ∑ t =1 ∞ Θ t X t + (where X + = max{0, X }) and max 1≤ k <∞ ∑ t =1 k Θ t X t , and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X 1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X 1 , X 2 ,... as independent and identically distributed positive random variables. If X 1 has a regularly varying tail of index -α, where α > 0, and if {Θ t , t ≥ 1} is a positive sequence of random variables independent of { X t }, then it is known - which can also be obtained from the sufficient conditions in this article - that, under some appropriate moment conditions on {Θ t , t ≥ 1}, X (∞) = ∑_ t =1 ∞ Θ t X t converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X (∞) has a regularly varying tail then does X 1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑ t =1 ∞ Θ t along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.

The convergence rate and asymptotic distribution of the bootstrap quantile variance estimator for importance sampling

Thu, 06 Sep 2012 14:15 EDT

Jingchen Liu, Xuan Yang

Source: Adv. in Appl. Probab., Volume 44, Number 3, 815--841.

Abstract:
Importance sampling is a widely used variance reduction technique to compute sample quantiles such as value at risk. The variance of the weighted sample quantile estimator is usually a difficult quantity to compute. In this paper we present the exact convergence rate and asymptotic distributions of the bootstrap variance estimators for quantiles of weighted empirical distributions. Under regularity conditions, we show that the bootstrap variance estimator is asymptotically normal and has relative standard deviation of order O ( n -1/4 ).

On exact sampling of nonnegative infinitely divisible random variables

Thu, 06 Sep 2012 14:15 EDT

Zhiyi Chi

Source: Adv. in Appl. Probab., Volume 44, Number 3, 842--873.

Abstract:
Nonnegative infinitely divisible (i.d.) random variables form an important class of random variables. However, when this type of random variable is specified via Lévy densities that have infinite integrals on (0, ∞), except for some special cases, exact sampling is unknown. We present a method that can sample a rather wide range of such i.d. random variables. A basic result is that, for any nonnegative i.d. random variable X with its Lévy density explicitly specified, if its distribution conditional on X ≤ r can be sampled exactly, where r > 0 is any fixed number, then X can be sampled exactly using rejection sampling, without knowing the explicit expression of the density of X . We show that variations of the result can be used to sample various nonnegative i.d. random variables.

A simple stochastic kinetic transport model

Thu, 06 Sep 2012 14:15 EDT

Michel Dekking, Derong Kong

Source: Adv. in Appl. Probab., Volume 44, Number 3, 874--885.

Abstract:
We introduce a discrete-time microscopic single-particle model for kinetic transport. The kinetics are modeled by a two-state Markov chain, and the transport is modeled by deterministic advection plus a random space step. The position of the particle after n time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to 0 linearly, we obtain a random variable S ( t ), which is closely connected to a well-known (deterministic) partial differential equation (PDE), reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed parts of the solute at time t do exist, and satisfy the PDEs.

On the optimal dividend strategy in a regime-switching diffusion model

Thu, 06 Sep 2012 14:15 EDT

Jiaqin Wei, Rongming Wang, Hailiang Yang

Source: Adv. in Appl. Probab., Volume 44, Number 3, 886--906.

Abstract:
In this paper we consider the optimal dividend strategy under the diffusion model with regime switching. In contrast to the classical risk theory, the dividends can only be paid at the arrival times of a Poisson process. By solving an auxiliary optimal problem we show that the optimal strategy is the modulated barrier strategy. The value function can be obtained by iteration or by solving the system of differential equations. We also provide a numerical example to illustrate the effects of the restriction on the timing of the payment of dividends.

The normalized graph cut and Cheeger constant: from discrete to continuous

Wed, 05 Dec 2012 09:10 EST

ERY ARIAS-CASTRO, BRUNO PELLETIER, PIERRE PUDLO

Source: Adv. in Appl. Probab., Volume 44, Number 4, 907--937.

Abstract:
Let M be a bounded domain of ∝ d with a smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M . By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M .

Set reconstruction by Voronoi cells

Wed, 05 Dec 2012 09:10 EST

M. Reitzner, E. Spodarev, D. Zaporozhets

Source: Adv. in Appl. Probab., Volume 44, Number 4, 938--953.

Abstract:
For a Borel set A and a homogeneous Poisson point process η in ∝ d of intensity λ>0, define the Poisson--Voronoi approximation A η of A as a union of all Voronoi cells with nuclei from η lying in A . If A has a finite volume and perimeter, we find an exact asymptotic of E Vol( A Δ A η ) as λ→∞, where Vol is the Lebesgue measure. Estimates for all moments of Vol( A η ) and Vol( A Δ A η ) together with their asymptotics for large λ are obtained as well.

Asymptotic properties of the approximate inverse estimator for directional distributions

Wed, 05 Dec 2012 09:10 EST

M. Riplinger, M. Spiess

Source: Adv. in Appl. Probab., Volume 44, Number 4, 954--976.

Abstract:
For stationary fiber processes, the estimation of the directional distribution is an important task. We consider a stereological approach, assuming that the intersection points of the process with a finite number of test hyperplanes can be observed in a bounded window. The intensity of these intersection processes is proportional to the cosine transform of the directional distribution. We use the approximate inverse method to invert the cosine transform and analyze asymptotic properties of the estimator in growing windows for Poisson line processes. We show almost-sure convergence of the estimator and derive Berry--Esseen bounds, including formulae for the variance.

Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process

Wed, 05 Dec 2012 09:10 EST

Joaquin Fontbona, Hélène Guérin, Florent Malrieu

Source: Adv. in Appl. Probab., Volume 44, Number 4, 977--994.

Abstract:
Motivated by stability questions on piecewise-deterministic Markov models of bacterial chemotaxis, we study the long-time behavior of a variant of the classic telegraph process having a nonconstant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both the velocity and position. Sharpness of the obtained convergence rate is discussed.

A strong law for the rate of growth of long latency periods in a cloud computing service

Wed, 05 Dec 2012 09:10 EST

Souvik Ghosh, Soumyadip Ghosh

Source: Adv. in Appl. Probab., Volume 44, Number 4, 995--1017.

Abstract:
Cloud-computing shares a common pool of resources across customers at a scale that is orders of magnitude larger than traditional multiuser systems. Constituent physical compute servers are allocated multiple `virtual machines' (VMs) to serve simultaneously. Each VM user should ideally be unaffected by others' demand. Naturally, this environment produces new challenges for the service providers in meeting customer expectations while extracting an efficient utilization from server resources. We study a new cloud service metric that measures prolonged latency or delay suffered by customers. We model the workload process of a cloud server and analyze the process as the customer population grows. The capacity required to ensure that the average workload does not exceed a threshold over long segments is characterized. This can be used by cloud operators to provide service guarantees on avoiding long durations of latency. As part of the analysis, we provide a uniform large deviation principle for collections of random variables that is of independent interest.

Scaling and multiscaling in financial series: a simple model

Wed, 05 Dec 2012 09:10 EST

ALESSANDRO ANDREOLI, FRANCESCO CARAVENNA, PAOLO DAI PRA, GUSTAVO POSTA

Source: Adv. in Appl. Probab., Volume 44, Number 4, 1018--1051.

Abstract:
We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate, and which captures some relevant stylized facts of financial assets, including scaling properties . In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation, and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average , with remarkable accuracy.

A hierarchical probability model of colon cancer

Wed, 05 Dec 2012 09:10 EST

Michael Kelly

Source: Adv. in Appl. Probab., Volume 44, Number 4, 1052--1083.

Abstract:
We consider a model of fixed size N = 2 l in which there are l generations of daughter cells and a stem cell. In each generation i there are 2 i -1 daughter cells. At each integral time unit the cells split so that the stem cell splits into a stem cell and generation 1 daughter cell and the generation i daughter cells become two cells of generation i +1. The last generation is removed from the population. A stem cell acquires first and second mutations at rates u 1 and u 2 , and a daughter cell acquires first and second mutations at rates v 1 and v 2 . We find the distribution for the time it takes to acquire two mutations as N goes to ∞ and the mutation rates go to 0. The mutation rates may tend to 0 at different speeds. We also find the distribution for the locations of the mutations. In particular, we determine whether or not the mutations occur on a stem cell and if not, at what generation in the daughter cells they occur. Several outcomes are possible, depending on how fast the rates go to 0. The model considered has been proposed by Komarova (2007) as a model for colon cancer.

Convex duality in mean-variance hedging under convex trading constraints

Wed, 05 Dec 2012 09:10 EST

Christoph Czichowsky, Martin Schweizer

Source: Adv. in Appl. Probab., Volume 44, Number 4, 1084--1112.

Abstract:
We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L 2 of the space of all gains from trade (i.e. the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad-hoc methods in specific frameworks.

Limit theorems for long-memory stochastic volatility models with infinite variance: partial sums and sample covariances

Wed, 05 Dec 2012 09:10 EST

RAFAŁ KULIK, PHILIPPE SOULIER

Source: Adv. in Appl. Probab., Volume 44, Number 4, 1113--1141.

Abstract:
In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

Asymptotics for weighted random sums

Wed, 05 Dec 2012 09:10 EST

MARIANA OLVERA-CRAVIOTO

Source: Adv. in Appl. Probab., Volume 44, Number 4, 1142--1172.

Abstract:
Let { X i } be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F ̄. Let ( N , C 1 , C 2 ,...) be a nonnegative random vector independent of the { X i } with N ∈ℕ∪ {∞}. We study the weighted random sum S N =∑ { i =1} N C i X i , and its maximum, M N =sup {1≤ k N +1 ∑ i =1 k C i X i . This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P( M N > x )∼ P( S N > x )∼ E[∑ i =1 N F ̄( x / C i )] as x →∞. When E[ X 1 ]>0 and the distribution of Z N =∑ i =1 N C i is also intermediate regularly varying, we obtain the asymptotics P( M N > x )∼ P( S N > x )∼ E[∑ i =1 N F ̄}( x / C i )] +P( Z N > x /E[ X 1 ]). For completeness, when the distribution of Z N is intermediate regularly varying and heavier than F ̄, we also obtain conditions under which the asymptotic relations P( M N > x) ∼ P( S N > x )∼ P( Z N > x / E[ X 1 ] hold.

Rare-event simulation of heavy-tailed random walks by sequential importance sampling and resampling

Wed, 05 Dec 2012 09:10 EST

HOCK PENG CHAN, SHAOJIE DENG, TZE-LEUNG LAI

Source: Adv. in Appl. Probab., Volume 44, Number 4, 1173--1196.

Abstract:
We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.

Full- and half-Gilbert tessellations with rectangular cells

Fri, 15 Mar 2013 09:29 EDT

James Burridge, Richard Cowan, Isaac Ma

Source: Adv. in Appl. Probab., Volume 45, Number 1, 1--19.

Abstract:
We investigate the ray-length distributions for two different rectangular versions of Gilbert's tessellation (see Gilbert (1967)). In the full rectangular version, lines extend either horizontally (east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a Poisson point process, each ray stopping when another ray is met. In the half rectangular version, east- and south-growing rays do not interact with west and north rays. For the half rectangular tessellation, we compute analytically, via recursion, a series expansion for the ray-length distribution, whilst, for the full rectangular version, we develop an accurate simulation technique, based in part on the stopping-set theory for Poisson processes (see Zuyev (1999)), to accomplish the same. We demonstrate the remarkable fact that plots of the two distributions appear to be identical when the intensity of seeds in the half model is twice that in the full model. In this paper we explore this coincidence, mindful of the fact that, for one model, our results are from a simulation (with inherent sampling error). We go on to develop further analytic theory for the half-Gilbert model using stopping-set ideas once again, with some novel features. Using our theory, we obtain exact expressions for the first and second moments of the ray length in the half-Gilbert model. For all practical purposes, these results can be applied to the full-Gilbert model—as much better approximations than those provided by Mackisack and Miles (1996).

Connectivity of random geometric graphs related to minimal spanning forests

Fri, 15 Mar 2013 09:29 EDT

C. Hirsch, D. Neuhäuser, V. Schmidt

Source: Adv. in Appl. Probab., Volume 45, Number 1, 20--36.

Abstract:
The almost-sure connectivity of the Euclidean minimal spanning forest MSF( X ) on a homogeneous Poisson point process X ⊂ ℝ d is an open problem for dimension d >2. We introduce a descending family of graphs ( G n ) n ≥2 that can be seen as approximations to the MSF in the sense that $MSF( X )=∩ n =2 ∞ G n ( X ). For n =2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity of G n ( X ) holds for all n ≥2, all dimensions d ≥2, and also point processes X more general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surely X does not admit generalized descending chains.

Fri, 15 Mar 2013 09:29 EDT

Jinghai Shao, Xiuping Wang

Source: Adv. in Appl. Probab., Volume 45, Number 1, 37--50.

Abstract:
Given two correlated Brownian motions ( X t ) t ≥ 0 and ( Y t ) t ≥ 0 with constant correlation coefficient, we give the upper and lower estimations of the probability ℙ(max 0 ≤ s ≤ t X s ≥ a , max 0 ≤ s ≤ t Y s ≥ b for any a , b , t >0 through explicit formulae. Our strategy is to establish a new reflection principle for two correlated Brownian motions, which can be viewed as an extension of the reflection principle for one-dimensional Brownian motion. Moreover, we also consider the nonexit probability for linear boundaries, i.e. ℙ ( X t ≤ a t + c ,( Y t ≤ b t + d , 0≤ t ≤ T ) for any constants a, b ≥0 and c,d, T >0.

Monotone policies and indexability for bidirectional restless bandits

Fri, 15 Mar 2013 09:29 EDT

K. D. Glazebrook, D. J. Hodge, C. Kirkbride

Source: Adv. in Appl. Probab., Volume 45, Number 1, 51--85.

Abstract:
Motivated by a wide range of applications, we consider a development of Whittle's restless bandit model in which project activation requires a state-dependent amount of a key resource, which is assumed to be available at a constant rate. As many projects may be activated at each decision epoch as resource availability allows. We seek a policy for project activation within resource constraints which minimises an aggregate cost rate for the system. Project indices derived from a Lagrangian relaxation of the original problem exist provided the structural requirement of indexability is met. Verification of this property and derivation of the related indices is greatly simplified when the solution of the Lagrangian relaxation has a state monotone structure for each constituent project. We demonstrate that this is indeed the case for a wide range of bidirectional projects in which the project state tends to move in a different direction when it is activated from that in which it moves when passive. This is natural in many application domains in which activation of a project ameliorates its condition, which otherwise tends to deteriorate or deplete. In some cases the state monotonicity required is related to the structure of state transitions, while in others it is also related to the nature of costs. Two numerical studies demonstrate the value of the ideas for the construction of policies for dynamic resource allocation, most especially in contexts which involve a large number of projects.

Error bounds for small jumps of Lvy processes

Fri, 15 Mar 2013 09:29 EDT

E. H. A. Dia

Source: Adv. in Appl. Probab., Volume 45, Number 1, 86--105.

Abstract:
The pricing of options in exponential Lévy models amounts to the computation of expectations of functionals of Lévy processes. In many situations, Monte Carlo methods are used. However, the simulation of a Lévy process with infinite Lévy measure generally requires either truncating or replacing the small jumps by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.

Characterizing heavy-tailed distributions induced by retransmissions

Fri, 15 Mar 2013 09:29 EDT

Predrag R. Jelenković, Jian Tan

Source: Adv. in Appl. Probab., Volume 45, Number 1, 106--138.

Abstract:
Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel availability dynamic is modeled as an independent and identically distributed sequence { A , A i } i ≥1 that is independent of L . During each period of time that the channel becomes available, say A i , we attempt to transmit the data unit. If L < A i , the transmission is considered successful; otherwise, we wait for the next available period A i +1 and attempt to retransmit the data from the beginning. We investigate the asymptotic properties of the number of retransmissions N and the total transmission time T until the data is successfully transmitted. In the context of studying the completion times in systems with failures where jobs restart from the beginning, it was first recognized by Fiorini, Sheahan and Lipsky (2005) and Sheahan, Lipsky, Fiorini and Asmussen (2006) that this model results in power-law and, in general, heavy-tailed delays. The main objective of this paper is to uncover the detailed structure of this class of heavy-tailed distributions induced by retransmissions. More precisely, we study how the functional relationship ℙ[ L > x ] -1 ≈ Φ (ℙ[ A > x ] -1 ) impacts the distributions of N and T ; the approximation `≈' will be appropriately defined in the paper based on the context. Depending on the growth rate of Φ(·), we discover several criticality points that separate classes of different functional behaviors of the distribution of N . For example, we show that if log(Φ( n )) is slowly varying then log(1/ℙ[ N > n ]) is essentially slowly varying as well. Interestingly, if log(Φ( n ))$ grows slower than e √(log n ) then we have the asymptotic equivalence log(ℙ[ N > n ]) ≈ - log(Φ( n ))$. However, if log(Φ( n )) grows faster than e √(log n ) , this asymptotic equivalence does not hold and admits a different functional form. Similarly, different types of distributional behavior are shown for moderately heavy tails (Weibull distributions) where log(ℙ[ N > n ]) ≈ -(logΦ( n )) 1/(β+1) , assuming that log \Φ( n ) ≈ n β , as well as the nearly exponential ones of the form log(ℙ[ N > n ]) ≈ - n /(log n ) 1/γ , γ>0, when Φ(·) grows faster than two exponential scales log log (Φ( n )) ≈ n γ .

Living on the multidimensional edge: seeking hidden risks using regular variation

Fri, 15 Mar 2013 09:29 EDT

Bikramjit Das, Abhimanyu Mitra, Sidney Resnick

Source: Adv. in Appl. Probab., Volume 45, Number 1, 139--163.

Abstract:
Multivariate regular variation plays a role in assessing tail risk in diverse applications such as finance, telecommunications, insurance, and environmental science. The classical theory, being based on an asymptotic model, sometimes leads to inaccurate and useless estimates of probabilities of joint tail regions. This problem can be partly ameliorated by using hidden regular variation (see Resnick (2002) and Mitra and Resnick (2011)). We offer a more flexible definition of hidden regular variation that provides improved risk estimates for a larger class of tail risk regions.

Bayesian quickest detection problems for some diffusion processes

Fri, 15 Mar 2013 09:29 EDT

Pavel V. Gapeev, Albert N. Shiryaev

Source: Adv. in Appl. Probab., Volume 45, Number 1, 164--185.

Abstract:
We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process with linear and exponential penalty costs for a detection delay. The optimal times of alarms are found as the first times at which the weighted likelihood ratios hit stochastic boundaries depending on the current observations. The proof is based on the reduction of the initial problems into appropriate three-dimensional optimal stopping problems and the analysis of the associated parabolic-type free-boundary problems. We provide closed-form estimates for the value functions and the boundaries, under certain nontrivial relations between the coefficients of the observable diffusion.

Asymptotics of Markov kernels and the tail chain

Fri, 15 Mar 2013 09:29 EDT

Sidney I. Resnick, David Zeber

Source: Adv. in Appl. Probab., Volume 45, Number 1, 186--213.

Abstract:
An asymptotic model for the extreme behavior of certain Markov chains is the `tail chain'. Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics, such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and nonextreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

Discrete-Time Semi-Markov Random Evolutions and their Applications

Fri, 15 Mar 2013 09:29 EDT

Nikolaos Limnios, Anatoliy Swishchuk

Source: Adv. in Appl. Probab., Volume 45, Number 1, 214--240.

Abstract:
In this paper we introduce discrete-time semi-Markov random evolutions (DTSMREs) and study asymptotic properties, namely, averaging, diffusion approximation, and diffusion approximation with equilibrium by the martingale weak convergence method. The controlled DTSMREs are introduced and Hamilton–Jacobi–Bellman equations are derived for them. The applications here concern the additive functionals (AFs), geometric Markov renewal chains (GMRCs), and dynamical systems (DSs) in discrete time. The rates of convergence in the limit theorems for DTSMREs and AFs, GMRCs, and DSs are also presented.

Finite- and infinite-time ruin probabilities with general stochastic investment return processes and bivariate upper tail independent and heavy-tailed claims

Fri, 15 Mar 2013 09:29 EDT

Fenglong Guo, Dingcheng Wang

Source: Adv. in Appl. Probab., Volume 45, Number 1, 241--273.

Abstract:
In this paper we investigate the asymptotic behaviors of the finite- and infinite-time ruin probabilities for a Poisson risk model with stochastic investment returns which constitute a general adapted càdlàg process and heavy-tailed claim sizes which are bivariate upper tail independent. The results of this paper show that the asymptotic ruin probabilities are dominated by the extreme of insurance risk but not by that of investment risk. As applications of the results, we discuss four special cases when the investment returns are determined by a fractional Brownian motion, an integrated Vasicek model, an integrated Cox–Ingersoll–Ross model, and the Heston model.

Loss systems with slow retrials in the Halfin–Whitt regime

Fri, 15 Mar 2013 09:29 EDT

F. Avram, A. J. E. M. Janssen, J. S. H. Van Leeuwaarden

Source: Adv. in Appl. Probab., Volume 45, Number 1, 274--294.

Abstract:
The Halfin–Whitt regime, or the quality-and-efficiency-driven (QED) regime, for multiserver systems refers to a situation with many servers, a critical load, and yet favorable system performance. We apply this regime to the classical multiserver loss system with slow retrials. We derive nondegenerate limiting expressions for the main steady-state performance measures, including the retrial rate and the blocking probability. It is shown that the economies of scale associated with the QED regime persist for systems with retrials, although in situations when the load becomes extremely critical the retrials cause deteriorated performance. Most of our results are obtained by a detailed analysis of Cohen's equation that defines the retrial rate in an implicit way. The limiting expressions are established by studying prelimit behavior and exploiting the connection between Cohen's equation and Mills' ratio for the Gaussian and Poisson distributions.