Showing papers in "Journal of Applied Probability in 2015"
TL;DR: In this article, the effect of changes in the scale parameters (λ 1, λ 2, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings was investigated.
Abstract: Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.
58 citations
TL;DR: In this paper, the authors considered point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν.
Abstract: In this paper we consider point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times τjlj of jumps with height lj (∑j=1rlj = k) under the condition N(t) = k for all these special processes is investigated in detail.
45 citations
TL;DR: In this article, the authors considered the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and studied the semidiscretization in time of the equation by an implicit Euler method.
Abstract: We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δt^γ) for any γ < ½. We also prove that the scheme converges uniformly in the strong L^p -sense but with no rate given.
42 citations
TL;DR: In this paper, the authors derived the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes and derived the normal approximation of the Parisian time.
Abstract: In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.
39 citations
TL;DR: In this paper, the authors analyzed the fractional Poisson process where the state probabilities p k ν k (t), t ≥ 0, are governed by time-fractional equations of order 0 < νk ≤ 1 depending on the number k of events that have occurred up to time t. They also introduced a different form of fractional state-dependent poisson process as a weighted sum of homogeneous Poisson processes.
Abstract: In this paper we analyse the fractional Poisson process where the state probabilities p k ν k (t), t ≥ 0, are governed by time-fractional equations of order 0 < ν k ≤ 1 depending on the number k of events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of p k ν k (t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on ν k differs from that constructed from the fractional state equations (in the case of ν k = ν, for all k, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.
29 citations
TL;DR: In this article, a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny is considered, where the unknown tree is modelled by a Yule process conditioned on n contemporary nodes.
Abstract: We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.
26 citations
TL;DR: The first-crossing time problem for the iterated Poisson process is finally tackled in the cases of a decreasing and constant boundary, where some closed-form results are provided and a linearly increasing boundary where an iterative procedure is proposed to compute the first-Crossing time density and survival functions.
Abstract: A compound Poisson process whose randomized time is an independent Poisson process is called a compound Poisson process with Poisson subordinator We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process The first-crossing time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing time density and survival functions
26 citations
TL;DR: In this paper, the authors studied the problem of determining the distribution and the moments of the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution.
Abstract: We study in this paper a generalized coupon collector problem, which consists in determining the distribution and the moments of the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we obtain expressions of the distribution and the moments of this time. We also prove that the almost-uniform distribution, for which all the non-null coupons have the same drawing probability, is the distribution which minimizes the expected time to get a fixed subset of distinct coupons. This optimization result is extended to the complementary distribution of that time when the full collection is considered, proving by the way this well-known conjecture. Finally, we propose a new conjecture which expresses the fact that the almost-uniform distribution should minimize the complementary distribution of the time needed to get any fixed number of distinct coupons.
25 citations
TL;DR: It follows that the number of protected nodes in a random recursive tree, upon proper scaling, converges in probability to a constant.
Abstract: We investigate protected nodes in random recursive trees. The exact mean of the number of such nodes is obtained by recurrence, and a linear asymptotic equivalent follows. A nonlinear recurrence for the variance shows that the variance grows linearly, too. It follows that the number of protected nodes in a random recursive tree, upon proper scaling, converges in probability to a constant.
24 citations
Journal Article•
TL;DR: This work introduces a modified version of the classical kernel and compares three different methods to simulate the Ginibre point process to discuss the most efficient one depending on the application at hand.
Abstract: The Ginibre point process is one of the main examples of deter- minantal point processes on the complex plane. It forms a recurring model in stochastic matrix theory as well as in pratical applications. However, this model has mostly been studied from a probabilistic point of view in the fields of stochastic matrices and determinantal point processes, and thus using the Ginibre process to model random phenomena is a topic which is for the most part unexplored. In order to obtain a determinantal point process more suited for simulation, we introduce a modified version of the classical kernel. Then, we compare three different methods to simulate the Ginibre point process and discuss the most efficient one depending on the application at hand.
24 citations
TL;DR: In this paper, the authors consider a dynamic metapopulation involving one large population of size N surrounded by colonies of size e N N, usually called peripheral isolates in ecology, where the main population, as well as the colonies, independently send propagules to found new colonies (emigration), and each colony independently, eventually merges with the main populations (fusion).
Abstract: We consider a dynamic metapopulation involving one large population of size N surrounded by colonies of size e N N, usually called peripheral isolates in ecology, where N → ∞ and e N → 0 in such a way that e N N → ∞. The main population, as well as the colonies, independently send propagules to found new colonies (emigration), and each colony independently, eventually merges with the main population (fusion). Our aim is to study the genealogical history of a finite number of lineages sampled at stationarity in such a metapopulation. We make assumptions on model parameters ensuring that the total outer population has size of the order of N and that each colony has a lifetime of the same order. We prove that under these assumptions, the scaling limit of the genealogical process of a finite sample is a censored coalescent where each lineage can be in one of two states: an inner lineage (belonging to the main population) or an outer lineage (belonging to some peripheral isolate). Lineages change state at constant rate and (only) inner lineages coalesce at constant rate per pair. This two-state censored coalescent is also shown to converge weakly, as the landscape dynamics accelerate, to a time-changed Kingman coalescent.
TL;DR: In this paper, the demand process is modeled as a Brownian motion and the optimal control policy is a generalized (s, S) policy consisting of a sequence of (si, Si ).
Abstract: In this paper we consider an inventory system with increasing concave ordering cost and average cost optimization criterion. The demand process is modeled as a Brownian motion. Porteus (1971) studied a discrete-time version of this problem and under the strong condition that the demand distribution belongs to the class of densities that are finite convolutions of uniform and/or exponential densities (note that normal density does not belong to this class), an optimal control policy is a generalized (s, S) policy consisting of a sequence of (si , Si ). Using a lower bound approach, we show that an optimal control policy for the Brownian inventory model is determined by a single pair (s, S).
TL;DR: In this article, the authors derived an exact asymptotic expansion (as u → ∞) of the Gaussian field with continuous sample paths and a correlation function, where the correlation function is Cov(X(s,t),X(0,0)) such that r(s,t) = 1 − |s| �1 − |t| �2 + o(|s|�1 + |t | �2 ), s,t → 0, with α1,α2 ∈ (0,2], and r(m,t
Abstract: Let {X(s,t) : s,t > 0} be a centered homogeneous Gaussian field with a.s. continuous sample paths and correlation function r(s,t) = Cov(X(s,t),X(0,0)) such that r(s,t) = 1 − |s| �1 − |t| �2 + o(|s| �1 + |t| �2 ), s,t → 0, with α1,α2 ∈ (0,2], and r(s,t) < 1 for (s,t) 6 (0,0). In this contribution we derive an exact asymptotic expansion (as u → ∞) of
TL;DR: In this article, the infinite-allele simple branching process of Griffiths and Pakes (1988) was extended to allow the offspring to change types and labels, and limit theorems were given for the growth of the number of labels of a specific type.
Abstract: We extend the infinite-allele simple branching process of Griffiths and Pakes (1988) allowing the offspring to change types and labels. The model is developed and limit theorems are given for the growth of the number of labels of a specific type. We also discuss the asymptotics of the frequency spectrum. Finally, we present an application of the model's use in tumorigenesis.
TL;DR: In this article, the ergodicity of subgeometric Markov chains is characterized in terms of the assumed drift and one-step minorisation conditions, and uniform bounds for these constants are needed for a family of Markov kernels.
Abstract: We provide explicit expressions for the constants involved in the characterisation of ergodicity of subgeometric Markov chains. The constants are determined in terms of those appearing in the assumed drift and one-step minorisation conditions. The results are fundamental for the study of some algorithms where uniform bounds for these constants are needed for a family of Markov kernels. Our results accommodate also some classes of inhomogeneous chains.
TL;DR: In this paper, the authors consider two types of drawdown time sequences depending on whether a historical running maximum is reset or not, and study the frequency rate of drawdowns, the Laplace transform of the $n$-th drawdown times, the distribution of the running maximum and the value process at the$n$th drawing time, as well as some other quantities of interest.
Abstract: Drawdowns measuring the decline in value from the historical running maxima
over a given period of time, are considered as extremal events from the
standpoint of risk management. To date, research on the topic has mainly focus
on the side of severity by studying the first drawdown over certain
pre-specified size. In this paper, we extend the discussion by investigating
the frequency of drawdowns, and some of their inherent characteristics. We
consider two types of drawdown time sequences depending on whether a historical
running maximum {is reset or not}. For each type, we study the frequency rate
of drawdowns, the Laplace transform of the $n$-th drawdown time, the
distribution of the running maximum and the value process at the $n$-th
drawdown time, as well as some other quantities of interest. Interesting
relationships between these two drawdown time sequences are also established.
Finally, insurance policies protecting against the risk of frequent drawdowns
are also proposed and priced.
TL;DR: In this paper, the authors define the reversed relevation transform as a dual to the relevation transformation and apply it to the lifetimes of parallel and series systems under suitably proportionality assumptions on the hazard rates.
Abstract: Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.
TL;DR: In this paper, the authors considered an M/M/c queue modified to allow both mass arrivals when the system is empty and the workload to be removed, and they derived the Laplace transformation of the transition probability for the absorptive queue.
Abstract: In this paper we consider an M/M/c queue modified to allow both mass arrivals when the system is empty and the workload to be removed. Properties of queues which terminate when the server becomes idle are firstly developed. Recurrence properties, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no mass exodus. All of these results are then generalized to allow for the removal of the entire workload. In particular, we obtain the Laplace transformation of the transition probability for the absorptive M/M/c queue.
TL;DR: In this paper, an extension of the standard Hawkes process by considering different exciting functions is proposed. But the main results are devoted to the asymptotic behavior of this extension of this process when unpredictable marks are considered.
Abstract: The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.
TL;DR: In this article, a modified version of the Ginibre point process (GPP) is introduced, which constitutes a determinantal point process more suited for certain applications, and its simulation is detailed.
Abstract: The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.
TL;DR: In this article, the authors derive families of approximate conditional sampling distributions for finite sites Λ- and Ξ-coalescents, and use them to obtain 'approximately optimal' importance sampling and approximate conditionals (PAC) algorithms.
Abstract: Full likelihood inference under Kingman's coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) methods have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general Λ- and Ξ-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites Λ- and Ξ-coalescents, and use them to obtain 'approximately optimal' IS and PAC algorithms for Λ-coalescents, yielding substantial gains in efficiency over existing methods.
TL;DR: In this paper, a simplified proof of the duality relation and answer most of the open questions posed in Krone (1999) are given. But they also fill in the details of an incomplete proof.
Abstract: In this paper, we continue the work started by Steve Krone on the two-stage contact process. We give a simplified proof of the duality relation and answer most of the open questions posed in Krone (1999). We also fill in the details of an incomplete proof.
TL;DR: Shot-noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory, and in the engineering sciences.
Abstract: Shot-noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory, and in the engineering sciences. In this paper we prove a large deviation principle for the sample-paths of a general class of multidimensional state-dependent Poisson shot-noise processes. The result covers previously known large deviation results for one-dimensional state-independent shot-noise processes with light tails. We use the weak convergence approach to large deviations, which reduces the proof to establishing the appropriate convergence of certain controlled versions of the original processes together with relevant results on existence and uniqueness.
TL;DR: In this article, the authors studied the optimal dividend payments for a company of limited liability whose cash reserves in the absence of dividends follow a Markov-modulated jump-diffusion process with positive drifts and negative exponential jumps, where parameters and discount rates are modulated by a finite-state irreducible Markov chain.
Abstract: In this paper we study the optimal dividend payments for a company of limited liability whose cash reserves in the absence of dividends follow a Markov-modulated jump-diffusion process with positive drifts and negative exponential jumps, where parameters and discount rates are modulated by a finite-state irreducible Markov chain. The main aim is to maximize the expected cumulative discounted dividend payments until bankruptcy time when cash reserves are nonpositive for the first time. We extend the results of Jiang and Pistorius [15] to our setup by proving that it is optimal to adopt a modulated barrier strategy at certain positive regime-dependent levels and that the value function can be explicitly characterized as the fixed point of a contraction.
TL;DR: The constructions not only help to deal with problems related to multivariate stochastic systems of lifetimes when joint defaults can occur with a nonzero probability, but even provide a copula maximizing the probability of joint default.
Abstract: We analyze copulas with a nontrivial singular component by using their Markov kernel representation. In particular, we provide existence results for copulas with a prescribed singular component. The constructions not only help to deal with problems related to multivariate stochastic systems of lifetimes when joint defaults can occur with a nonzero probability, but even provide a copula maximizing the probability of joint default.
TL;DR: The existence of an almost surely asymptotic degree distribution with stretched exponential decay was shown in this paper, where it was shown that the proportion of vertices of degree d tends to some positive number c(d) > 0 almost surely, as the number of steps goes to infinity.
Abstract: We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number c(d) > 0 almost surely as the number of steps goes to infinity, and c(d) similar to (e pi)(1/2)d(1/4)e(-2)root d holds as d -> infinity.
TL;DR: In this article, the authors developed a prelimit analysis of performance measures for importance sampling schemes related to small noise diffusion processes, and characterized the second moment of the corresponding estimator as the solution to a partial differential equation, which they analyzed via a full asymptotic expansion with respect to the size of the noise and obtained a precise statement on its accuracy.
Abstract: In this paper we develop a prelimit analysis of performance measures for importance sampling schemes related to small noise diffusion processes. In importance sampling the performance of any change of measure is characterized by its second moment. For a given change of measure, we characterize the second moment of the corresponding estimator as the solution to a partial differential equation, which we analyze via a full asymptotic expansion with respect to the size of the noise and obtain a precise statement on its accuracy. The main correction term to the decay rate of the second moment solves a transport equation that can be solved explicitly. The asymptotic expansion that we obtain identifies the source of possible poor performance of nevertheless asymptotically optimal importance sampling schemes and allows for a more accurate comparison among competing importance sampling schemes.
TL;DR: The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal and Katona (1968).
Abstract: We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.
TL;DR: In this article, the authors take a fresh look at the classical problem of runs in a sequence of independent and identically distributed coin tosses and derive a general identity/recursion which can be used to compute (joint) distributions of functionals of run types.
Abstract: We take a fresh look at the classical problem of runs in a sequence of independent and identically distributed coin tosses and derive a general identity/recursion which can be used to compute (joint) distributions of functionals of run types. This generalizes and unifies already existing approaches. We give several examples, derive asymptotics, and pose some further questions.