bluuebunny/crossref_metadata_2025 huggingface dataset:
Compound Poisson limits for household epidemics
Authors: Neal Peter
Published: July 2016

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.
Differentiation of some functionals of risk processes, and optimal reserve allocation
Authors: Loisel Stéphane
Published: July 2016

For general risk processes, we introduce and study the expected time-integrated negative part of the process on a fixed time interval. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below 0, studied by Dos Reis, and the probability of ruin. We carry out differentiation of other functionals of one-dimensional and multidimensional risk processes with respect to the initial reserve level. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve that minimizes one of these risk measures, illustrate the variety of fields of application and the benefits deriving from an efficient and effective use of such tools.
Multiperiod conditional distribution functions for conditionally normal GARCH(1, 1) models
Authors: Brummelhuis Raymond , Guégan Dominique
Published: July 2016

We study the asymptotic tail behavior of the conditional probability distributions of r

t+k
and r

t+1+⋯+r

t+k
when (r

t
)
t∈ℕ is a GARCH(1, 1) process. As an application, we examine the relation between the extreme lower quantiles of these random variables.
On the expected time to ruin and the expected dividends when dividends are paid while the surplus is above a constant barrier
Authors: Frostig Esther
Published: July 2016

We study the expected time to ruin in a risk process in which dividends are paid when the surplus is above the barrier. We consider the case in which the dividend rate is smaller than the premium rate. We obtain results for the classical compound Poisson risk process with phase-type claim size. When the ruin probability is 1, we derive the expected time to ruin and the expected dividends paid. When the ruin probability is less than 1, these quantities are derived conditioning on the event that ruin occurs.
Minimum dynamic discrimination information models
Authors: Asadi Majid , Ebrahimi Nader , Hamedani G. G. , Soofi Ehsan S.
Published: July 2016

In this paper, we introduce the minimum dynamic discrimination information (MDDI) approach to probability modeling. The MDDI model relative to a given distribution G is that which has least Kullback-Leibler information discrepancy relative to G, among all distributions satisfying some information constraints given in terms of residual moment inequalities, residual moment growth inequalities, or hazard rate growth inequalities. Our results lead to MDDI characterizations of many well-known lifetime models and to the development of some new models. Dynamic information constraints that characterize these models are tabulated. A result for characterizing distributions based on dynamic Rényi information divergence is also given.
Recursive filters for partially observable finite Markov chains
Authors: Ledoux James
Published: July 2016

In this note, we consider discrete-time finite Markov chains and assume that they are only partly observed. We obtain finite-dimensional normalized filters for basic statistics associated with such processes. Recursive equations for these filters are derived by means of simple computations involving conditional expectations. An application to the estimation of parameters of the so-called discrete-time batch Markovian arrival process is outlined.
On the existence of uni-instantaneous Q-processes with a given finite μ-invariant measure
Authors: Gray Brenton , Pollett Phil , Zhang Hanjun
Published: July 2016

Let S be a countable set and let Q = (q

ij
, i, j ∈ S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number μ ≥ 0 and a strictly positive probability measure m = (m

j
, j ∈ S) such that ∑
i∈S

m

i

q

ij
= −μ
m

j
, j ≠ b. We prove that there exists a Q-process P(t) = (p

ij
(t), i, j ∈ S) for which m is a μ-invariant measure, that is ∑
i∈S

m

i

p

ij
(t) = e−μ
t

m

j
, j∈S. We illustrate our results with reference to the Kolmogorov ‘K1’ chain and a birth-death process with catastrophes and instantaneous resurrection.
A stochastic SIS infection model incorporating indirect transmission
Authors: Clancy Damian
Published: July 2016

We describe a stochastic susceptible–infective–susceptible (SIS) model for transmission of infectious disease through a population, incorporating both direct host–host transmission and indirect transmission via free-living infectious stages (e.g. environmental bacteria). Existence of a quasi-stationary distribution conditional upon nonextinction of infection is established. A bivariate Ornstein–Uhlenbeck approximation is used to investigate the long-term behaviour of the process conditional upon nonextinction of infection. We show that indirect transmission leads to lower variability in the number of infected hosts present in quasi-stationarity and, consequently, to a greater tendency of infection to persist, compared with a model with direct transmission only and the same average individual infectivity. Some numerical work illustrating these results is presented.
Tax problems in the undiscounted case
Authors: Whittle P.
Published: July 2016

The aim of this paper is to evaluate the performance of the optimal policy (the Gittins index policy) for open tax problems of the type considered by Klimov in the undiscounted limit. In this limit, the state-dependent part of the cost is linear in the state occupation numbers for the multi-armed bandit, but is quadratic for the tax problem. The discussion of the passage to the limit for the tax problem is believed to be largely new; the principal novelty is our evaluation of the matrix of the quadratic form. These results are confirmed by a dynamic programming analysis, which also suggests how the optimal policy should be modified when resources can be freely deployed only within workstations, rather than system-wide.
A complete solution for the optimal stochastic scheduling of a two-stage tandem queue with two flexible servers
Authors: Schiefermayr Klaus , Weichbold Josef
Published: July 2016

We consider a two-stage tandem queue with two parallel servers and two queues. We assume that initially all jobs are present and that no further arrivals take place at any time. The two servers are identical and can serve both types of job. The processing times are exponentially distributed. After being served, a job of queue 1 joins queue 2, whereas a job of queue 2 leaves the system. Holding costs per job and per unit time are incurred if there are jobs holding in the system. Our goal is to find the optimal strategy that minimizes the expected total holding costs until the system is cleared. We give a complete solution for the optimal control of all possible parameters (costs and service times), especially for those parameter regions in which the optimal control depends on how many jobs are present in the two queues.
Conditional tail expectations for multivariate phase-type distributions
Authors: Cai Jun , Li Haijun
Published: July 2016

The conditional tail expectation in risk analysis describes the expected amount of risk that can be experienced given that a potential risk exceeds a threshold value, and provides an important measure of right-tail risk. In this paper, we study the convolution and extreme values of dependent risks that follow a multivariate phase-type distribution, and derive explicit formulae for several conditional tail expectations of the convolution and extreme values for such dependent risks. Utilizing the underlying Markovian property of these distributions, our method not only provides structural insight, but also yields some new distributional properties of multivariate phase-type distributions.
Stopping at the maximum of geometric Brownian motion when signals are received
Authors: Guo X. , Liu J.
Published: July 2016

Consider a geometric Brownian motion X

t
(ω) with drift. Suppose that there is an independent source that sends signals at random times τ
1 < τ
2 < ⋯. Upon receiving each signal, a decision has to be made as to whether to stop or to continue. Stopping at time τ will bring a reward S

τ
, where S

t
= max(max0≤u≤t

X

u
, s) for some constant s ≥ X
0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−r
τ

i

S

τ

i

| X
0 = x, S
0 = s], where r is a discount factor. This problem can be viewed as a randomized version of the Bermudan look-back option pricing problem. In this paper, we derive explicit solutions to this optimal stopping problem, assuming that signal arrival is a Poisson process with parameter λ. Optimal stopping rules are differentiated by the frequency of the signal process. Specifically, there exists a threshold λ* such that if λ>λ*, the optimal stopping problem is solved via the standard formulation of a ‘free boundary’ problem and the optimal stopping time τ
* is governed by a threshold a
* such that τ
* = inf{τ

n
: X

τ

n

≤a
*
S

τ

n

}. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ
* = τ
1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.
A sufficient condition for the existence of an invariant probability measure for Markov processes
Authors: Costa O. L. V. , Dufour F.
Published: July 2016

In this paper, it is shown that the Foster-Lyapunov criterion is sufficient to ensure the existence of an invariant probability measure for both discrete- and continuous-time Markov processes without any additional hypotheses (such as irreducibility).
Sensitivity and convergence of uniformly ergodic Markov chains
Authors: Mitrophanov A. Yu.
Published: July 2016

For uniformly ergodic Markov chains, we obtain new perturbation bounds which relate the sensitivity of the chain under perturbation to its rate of convergence to stationarity. In particular, we derive sensitivity bounds in terms of the ergodicity coefficient of the iterated transition kernel, which improve upon the bounds obtained by other authors. We discuss convergence bounds that hold in the case of finite state space, and consider numerical examples to compare the accuracy of different perturbation bounds.
Simple derivations of properties of counting processes associated with Markov renewal processes
Authors: Ball Frank , Milne Robin K.
Published: July 2016

A simple, widely applicable method is described for determining factorial moments of N̂

t
, the number of occurrences in (0,t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t → ∞. The factorial moment formulae combine to yield an expression for the probability generating function of N̂

t
, and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.
How to estimate the rate function of a cumulative process
Authors: Duffy Ken , Metcalfe Anthony P.
Published: July 2016

Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate function J(⋅,⋅) and whose cumulative process satisfies the LDP with rate function I(⋅). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of a cumulative renewal process it is demonstrated that this approach is favourable to a more direct method, as it ensures that the laws of the estimates converge weakly to a Dirac measure at I.
A uniformly convergent adaptive particle filter
Authors: Papavasiliou Anastasia
Published: July 2016

Particle filters are Monte Carlo methods that aim to approximate the optimal filter of a partially observed Markov chain. In this paper, we study the case in which the transition kernel of the Markov chain depends on unknown parameters: we construct a particle filter for the simultaneous estimation of the parameter and the partially observed Markov chain (adaptive estimation) and we prove the convergence of this filter to the correct optimal filter, as time and the number of particles go to infinity. The filter presented here generalizes Del Moral's Monte Carlo particle filter.
Cut-off and hitting times of a sample of Ornstein-Uhlenbeck processes and its average
Authors: Lachaud B.
Published: July 2016

A cut-off phenomenon is shown to occur in a sample of n independent, identically distributed Ornstein-Uhlenbeck processes and its average. Their distributions stay far from equilibrium before a certain O(log(n)) time, and converge exponentially fast after. Precise estimates show that the total variation distance drops from almost 1 to almost 0 over an interval of time of length O(1) around log(n)/(2α), where α is the viscosity coefficient of the sampled process. The distribution of the hitting time of 0 by the average of the sample is computed. As n tends to infinity, the hitting time becomes concentrated around the cut-off instant, and its tails match the estimates given for the total variation distance.
Optimality of the 2-CUSUM drift equalizer rules for detecting two-sided alternatives in the Brownian motion model
Authors: Hadjiliadis Olympia
Published: July 2016

This work employs the Brownian motion model in which observations are taken sequentially. The objective is to detect a two-sided change in the constant drift by means of a stopping rule. As a performance measure, an extended Lorden criterion is used. The goal is to minimize the worst-case detection delay subject to a constraint in the frequency of false alarms. In a companion paper, attention is drawn to a first category of 2-CUSUM rules for which the harmonic mean rule holds. It is further seen that a special class of 2-CUSUM stopping rules within this category, called drift equalizer rules, perform strictly better than non-equalizer rules, according to this specific performance measure.
Le Her with s Suits and d Denominations
Authors: Vanniasegaram Sithparran
Published: July 2016

In 2002, Benjamin and Goldman gave a complete solution to a variant of the two-player card game Le Her. We extend their result by giving optimal strategies for the authentic version played with a deck consisting of arbitrary numbers of suits and denominations. Additionally, we show that the player who has the advantage in the game when one standard deck is used does not have the advantage if nineteen or more standard decks are used.
Positive Dependence and Weak Convergence
Authors: Colangelo A. , Müller A. , Scarsini M.
Published: July 2016

A more general definition of MTP2 (multivariate total positivity of order 2) probability measure is given, without assuming the existence of a density. Under this definition the class of MTP2 measures is proved to be closed under weak convergence. Characterizations of the MTP2 property are proved under this more general definition. Then a precise definition of conditionally increasing measure is provided, and closure under weak convergence of the class of conditionally increasing measures is proved. As an application we investigate MTP2 properties of stationary distributions of Markov chains, which are of interest in actuarial science.
Several Types of Ergodicity for M/G/1-Type Markov Chains and Markov Processes
Authors: Liu Yuanyuan , Hou Zhenting
Published: July 2016

In this paper we study polynomial and geometric (exponential) ergodicity for M/G/1-type Markov chains and Markov processes. First, practical criteria for M/G/1-type Markov chains are obtained by analyzing the generating function of the first return probability to level 0. Then the corresponding criteria for M/G/1-type Markov processes are given, using their h-approximation chains. Our method yields the radius of convergence of the generating function of the first return probability, which is very important in obtaining explicit bounds on geometric (exponential) convergence rates. Our results are illustrated, in the final section, in some examples.
On Maxima and Ladder Processes for a Dense Class of Lévy Process
Authors: Pistorius Martijn
Published: July 2016

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.
Limit Theorems for Moving Averages with Random Coefficients and Heavy-Tailed Noise
Authors: Kulik Rafał
Published: July 2016

We consider a stationary moving average process with random coefficients, , generated by an array, {C

t,k
, t ∈ Z, k ≥ 0}, of random variables and a heavy-tailed sequence, {Z

t
, t ∈ Z}. We analyze the limit behavior using a point process analysis. As applications of our results we compare the limiting behavior of the moving average process with random coefficients with that of a standard MA(∞) process.
On the Waiting Time to Escape
Authors: Serra Maria Conceição
Published: July 2016

The mathematical model we consider here is a decomposable Galton-Watson process with individuals of two types, 0 and 1. Individuals of type 0 are supercritical and can only produce individuals of type 0, whereas individuals of type 1 are subcritical and can produce individuals of both types. The aim of this paper is to study the properties of the waiting time to escape, i.e. the time it takes to produce a type-0 individual that escapes extinction when the process starts with a type-1 individual. With a view towards applications, we provide examples of populations in biological and medical contexts that can be suitably modeled by such processes.
On-line selection of an acceptable pair
Authors: Preater J.
Published: July 2016

A sequence of objects with independent, identically distributed qualities is presented to a selector who must choose two on-line, i.e. without anticipation or recall. The selector's aim is to obtain a satisfactory pair as quickly as possible. Two versions of the problem are considered, and optimal selection rules are derived and compared. An investigation is also made of a heuristic rule suitable for a selector who has no prior knowledge of the nature of the object sequence.
Ergodic properties and ergodic decompositions of continuous-time Markov processes
Authors: Costa O. L. V. , Dufour F.
Published: July 2016

In this paper we obtain some ergodic properties and ergodic decompositions of a continuous-time, Borel right Markov process taking values in a locally compact and separable metric space. Initially, we assume that an invariant probability measure (IPM) μ exists for the process and, without making any further assumptions on the transition kernel, obtain some characterization results for the convergence of the expected occupation measure to a limit kernel. Under the same assumption, we present the so-called Yosida decomposition. Next, instead of assuming the existence of an IPM, we assume that the Markov process satisfies a certain condition, named theT'-condition. Under this condition it is shown that the Foster-Lyapunov criterion is necessary and sufficient for the existence of an IPM and that the process admits a Doeblin decomposition. Furthermore, it is shown that in this case the set of ergodic probability measures is countable and that every probability measure for the Markov process is nonsingular with respect to the transition kernel.
Imperfect maintenance in a generalized competing risks framework
Authors: Doyen Laurent , Gaudoin Olivier
Published: July 2016

In this paper we present a general framework for the modelling of the process of corrective and condition-based preventive maintenance actions for complex repairable systems. A new class of models is proposed, the generalized virtual age models. On the one hand, these models generalize Kijima's virtual age models to the case where both preventive and corrective maintenances are present. On the other hand, they generalize the usual competing risks models to imperfect maintenance actions which do not renew the system. A generalized virtual age model is defined by both a sequence of effective ages which characterizes the effects of both types of maintenance according to a classical virtual age model, and a usual competing risks model which characterizes the dependency between the two types of maintenance. Several particular cases of the general model are derived.
Two-person red-and-black games with bet-dependent win probability functions
Authors: Chen May-Ru , Hsiau Shoou-Ren
Published: July 2016

In this paper a two-person red-and-black game is investigated. We suppose that, at every stage of the game, player I's win probability, f, is a function of the ratio of his bet to the sum of both players' bets. Two results are given: (i) if f is convex then a bold strategy is optimal for player I when player II plays timidly; and (ii) if f satisfies f(s)f(t) ≤ f(st) then a timid strategy is optimal for player II when player I plays boldly. These two results extend two formulations of red-and-black games proposed by Pontiggia (2005), and also provide a sufficient condition to ensure that the profile (bold, timid) is the unique Nash equilibrium for players I and II. Finally, we give a counterexample to Pontiggia's conjecture about a proportional N-person red-and-black game.
Critical sizing of LRU caches with dependent requests
Authors: Jelenković Predrag R. , Radovanović Ana , Squillante Mark S.
Published: July 2016

It was recently proved by Jelenković and Radovanović (2004) that the least-recently-used (LRU) caching policy, in the presence of semi-Markov-modulated requests that have a generalized Zipf's law popularity distribution, is asymptotically insensitive to the correlation in the request process. However, since the previous result is asymptotic, it remains unclear how small the cache size can become while still retaining the preceding insensitivity property. In this paper, assuming that requests are generated by a nearly completely decomposable Markov-modulated process, we characterize the critical cache size below which the dependency of requests dominates the cache performance. This critical cache size is small relative to the dynamics of the modulating process, and in fact is sublinear with respect to the sojourn times of the modulated chain that determines the dependence structure.
Rendezvous Search with Revealed Information: Applications to the Line
Authors: Alpern Steve
Published: July 2016

The symmetric rendezvous problem on a network Q asks how two players, forced to use the same mixed strategy, can minimize their expected meeting time, starting from a known initial distribution on the nodes of Q. This minimum is called the (symmetric) ‘rendezvous value’ of Q. Traditionally, the players are assumed to receive no information while playing the game. We consider the effect on rendezvous times of giving the players some information about past actions and chance moves, enabling each of them to apply Bayesian updates to improve his knowledge of the other's whereabouts. This technique can be used to give lower bounds on the rendezvous times of the original game (without any revealed information). We consider the case in which they are placed a known distance apart on the line graph Q (known as ‘symmetric rendezvous on the line’). Our approach is to concentrate on a general analysis of the effect of revelations, rather than compute the best bounds possible with our technique.
A Class of Multidimensional Q-Processes
Authors: Wu Bo , Zhang Yu-Hui
Published: July 2016

In this paper we present some necessary conditions for the uniqueness, recurrence, and ergodicity of a class of multidimensionalQ-processes, using the dual Yan-Chen comparison method. Then the coupling method is used to study the multidimensional processes in a specific space. As applications, three models of particle systems are illustrated.
A Stochastic Ordering Property for Leaky Bucket Regulated Flows in Packet Networks
Authors: Guillemin Fabrice M. , Mazumdar Ravi R. , Rosenberg Catherine P. , Ying Yu
Published: July 2016

We show in this paper that if a stationary traffic source is regulated by a leaky bucket with leak rate ρ and bucket size σ, then the amount of information generated in successive time intervals is dominated, in the increasing convex ordering sense, by that of a Poisson arrival process with rate ρ/σ, with each arrival bringing an amount of information equal to σ. By exploiting this property, we then show that the mean value in the stationary regime of the content of a buffer drained at constant rate and fed with the superposition of regulated flows is less than the mean value of the same buffer fed with an adequate Poisson process, whose characteristics depend upon the regulated input flows.
A Stability Property of Stochastic Vibration
Authors: Elshamy M.
Published: July 2016

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.
A Note on Extinction Times for the General Birth, Death and Catastrophe Process
Authors: Pollett Phil , Zhang Hanjun , Cairns Benjamin J.
Published: July 2016

We consider a birth, death and catastrophe process where the transition rates are allowed to depend on the population size. We obtain an explicit expression for the expected time to extinction, which is valid in all cases where extinction occurs with probability 1.
A Stochastic Maximin Fixed-Point Equation Related to Game Tree Evaluation
Authors: Alsmeyer Gerold , Meiners Matthias
Published: July 2016

After suitable normalization the asymptotic root valueWof a minimax game tree of orderb≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval ofRappears to be a particular solution of the stochastic maximin fixed-point equationWξ max1≤i≤bmin1≤j≤bWi,j, whereWi,jare independent copies ofWanddenotes equality in law. Moreover, ξ=g'(α) > 1, whereg(x) := (1 − (1 −x)b)band α denotes the unique fixed point ofgin (0, 1). This equation, which takes the formF(t) =g(F(t/ξ)) in terms of the distribution functionFofW, is studied in the present paper for a reasonably extended class of functionsgso as to encompass more general stochastic maximin equations as well. A complete description of the set of solutionsFis provided followed by a discussion of additional properties such as continuity, differentiability, or existence of moments. Based on these results, it is further shown that the particular solution mentioned above stands out among all other ones in that its distribution function is the restriction of an entire function to the real line. This extends recent work of Ali Khan, Devroye and Neininger (2005). A connection with another class of stochastic fixed-point equations for weighted minima and maxima is also discussed.
Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution
Authors: Barbe Ph. , McCormick W. P. , Zhang C.
Published: July 2016

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.
Ruin Probability for the Integrated Gaussian Process with Force of Interest
Authors: He Xiaoxia , Hu Yijun
Published: July 2016

In this paper we obtain the exact asymptotics of the ruin probability for the integrated Gaussian process with force of interest. The results obtained are consistent with those obtained for the case in which there is no force of interest.
Asymptotic Behavior of k-Word Matches Between two Uniformly Distributed Sequences
Authors: Kantorovitz M. R. , Booth H. S. , Burden C. J. , Wilson S. R.
Published: July 2016

Given two sequences of length n over a finite alphabet A of size |A| = d, the D
2 statistic is the number of k-letter word matches between the two sequences. This statistic is used in bioinformatics for EST sequence database searches. Under the assumption of independent and identically distributed letters in the sequences, Lippert, Huang and Waterman (2002) raised questions about the asymptotic behavior of D
2 when the alphabet is uniformly distributed. They expressed a concern that the commonly assumed normality may create errors in estimating significance. In this paper we answer those questions. Using Stein's method, we show that, for large enough k, the D
2 statistic is approximately normal as n gets large. When k = 1, we prove that, for large enough d, the D
2 statistic is approximately normal as n gets large. We also give a formula for the variance of D
2 in the uniform case.
Reversed Preservation Properties for Series and Parallel Systems
Authors: Belzunce Félix , Martínez-Puertas Helena , Ruiz José M.
Published: July 2016

Recently Li and Yam (2005) studied which ageing properties for series and parallel systems are inherited for the components. In this paper we provide new results for the increasing convex and concave orders, the increasing mean residual life (IMRL), decreasing failure rate (DFR), the new worse than used in expectation (NWUE), the increasing failure rate in average (IFRA), the decreasing failure rate in average (DFRA), and the new better than used in the convex order (NBUC) ageing classes.
A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics
Authors: Feng Shui , Wang Feng-Yu
Published: July 2016

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := {
x
∈ [0, 1]

N

: ∑
i≥1
x

i
= 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).
A Unifying Conservation Law for Single-Server Queues
Authors: Ayesta Urtzi
Published: July 2016

We develop a conservation law for a multi-class GI/GI/1 queue operating under a general work-conserving scheduling discipline. For single-class single-server queues, conservation laws have been obtained for both nonanticipating and anticipating disciplines with general service time distributions. For multi-class single-server queues, conservation laws have been obtained for (i) nonanticipating disciplines with exponential service time distributions and (ii) nonpreemptive nonanticipating disciplines with general service time distributions. The unifying conservation law we develop generalizes already existing conservation laws. In addition, it covers popular nonanticipating multi-class time-sharing disciplines such as discriminatory processor sharing (DPS) and generalized processor sharing (GPS) with general service time distributions. As an application, we show that the unifying conservation law can be used to compare the expected unconditional response time under two scheduling disciplines.
Tail Asymptotics for the Queue Size Distribution in an M/G/1 Retrial Queue
Authors: Kim Jerim , Kim Bara , Ko Sung-Seok
Published: July 2016

We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.
Ordering of Optimal Portfolio Allocations in a Model with a Mixture of Fundamental Risks
Authors: Cheung Ka Chun , Yang Hailiang
Published: July 2016

In this paper we study a single-period optimal portfolio problem in which the aim of the investor is to maximize the expected utility. We assume that the return of every security in the market is a mixture of some common underlying source of risks. A sufficient condition to order the optimal allocations is obtained, and it is shown that several models studied in the literature before are special cases of the proposed model. In the course of the analysis concepts in stochastic orders are employed, and a new characterization of the likelihood ratio order is obtained.
Wiener-Hopf Factorization for Lévy Processes Having Positive Jumps with Rational Transforms
Authors: Lewis Alan L. , Mordecki Ernesto
Published: July 2016

We show that the positive Wiener-Hopf factor of a Lévy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation. Based on this, we give the closed form of the ruin probability for a Lévy process, with completely arbitrary negatively distributed jumps, and finite intensity positive jumps with a distribution characterized by a rational Fourier transform. We also obtain results for the ladder process and its Laplace exponent. A key role is played by the analytic properties of the characteristic exponent of the process and by a Baxter-Donsker-type formula for the positive factor that we derive.
Weak Local Linear Discretizations for Stochastic Differential Equations with Jumps
Authors: Carbonell F. , Jimenez J. C.
Published: July 2016

Weak local linear approximations have played a prominent role in the construction of effective inference methods and numerical integrators for stochastic differential equations. In this note two weak local linear approximations for stochastic differential equations with jumps are introduced as a generalization of previous ones. Their respective order of convergence is obtained as well.
Continuous Mixtures with Bathtub-Shaped Failure Rates
Authors: Block Henry W. , Li Yulin , Savits Thomas H. , Wang Jie
Published: July 2016

The failure rate of a mixture of even the most standard distributions used in reliability can have a complicated shape. However, failure rates of mixtures of two carefully selected distributions will have the well-known bathtub shape. Here we show that mixtures of whole families of distribtions can have a bathtub-shaped failure rate.
Polynomial-Rate Convergence to the Stationary State for the Continuum-Time Limit of the Minority Game
Authors: Ortisi Matteo
Published: July 2016

In this paper we show that the continuum-time version of the minority game satisfies the criteria for the application of a theorem on the existence of an invariant measure. We consider the special case of a game with a ‘sufficiently’ asymmetric initial condition, where the number of possible choices for each individual is S = 2 and Γ < +∞. An upper bound for the asymptotic behavior, as the number of agents grows to infinity, of the waiting time for reaching the stationary state is then obtained.
Equilibrium Selection in Evolutionary Games with Imperfect Monitoring
Authors: Chen Hsiao-Chi , Chow Yunshyong
Published: July 2016

In this paper we analyze players' long-run behavior in evolutionary coordination games with imperfect monitoring in a large population. Players can observe signals corresponding to other players' unseen actions and use the proposed simple or maximum likelihood estimation algorithm to extract information from the signals. In the simple learning process we find conditions for the risk-dominant and the non-risk-dominant equilibria to emerge alone in the long run. Furthermore, we find that the two equilibria can coexist in the long run. In contrast, the coexistence of the two equilibria is the only limit distribution under the maximum likelihood estimation learning algorithm. We also analyze the long-run equilibria of other 2x2 symmetric games under imperfect monitoring.
Poisson Approximation for the Number of Repeats in a Stationary Markov Chain
Authors: Touyar Narjiss , Schbath Sophie , Cellier Dominique , Dauchel Hélène
Published: July 2016

Detection of repeated sequences within complete genomes is a powerful tool to help understanding genome dynamics and species evolutionary history. To distinguish significant repeats from those that can be obtained just by chance, statistical methods have to be developed. In this paper we show that the distribution of the number of long repeats in long sequences generated by stationary Markov chains can be approximated by a Poisson distribution with explicit parameter. Thanks to the Chen-Stein method we provide a bound for the approximation error; this bound converges to 0 as soon as the length n of the sequence tends to ∞ and the length t of the repeats satisfies n
2ρ
t
= O(1) for some 0 < ρ < 1. Using this Poisson approximation, p-values can then be easily calculated to determine if a given genome is significantly enriched in repeats of length t.
A Note on Conditioning and Stochastic Domination for Order Statistics
Authors: Dubhashi Devdatt , Häggström Olle
Published: July 2016

For an order statistic (X
1:n
,…,X

n:n
) of a collection of independent but not necessarily identically distributed random variables, and any i ∈ {1,…,n}, the conditional distribution of (X

i+1:n
,…,X

n:n
) given X

i:n
> s is shown to be stochastically increasing in s. This answers a question by Hu and Xie (2006).
Two Remarks on Blackwell's Theorem
Authors: Lehrer Ehud , Shmaya Eran
Published: July 2016

In a decision problem with uncertainty a decision maker receives partial information about the actual state via an information structure. After receiving a signal, he is allowed to withdraw and gets zero profit. We say that one structure is better than another when a withdrawal option exists if it may never happen that one structure guarantees a positive profit while the other structure guarantees only zero profit. This order between information structures is characterized in terms that are different from those used by Blackwell's comparison of experiments. We also treat the case of a malevolent nature that chooses a state in an adverse manner. It turns out that Blackwell's classical characterization also holds in this case.
The Generalised Coupon Collector Problem
Authors: Neal Peter
Published: July 2016

Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distribution X with the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons n → ∞ under general assumptions upon the asymptotic distribution of X. The results are proved by studying the total number of coupons, W(t), not collected in t draws from the population and noting that P(Y ≤ t) = P(W(t) = 0). Two normalizations of Y are considered, the choice of normalization depending upon whether or not a suitable Poisson limit exists for W(t). Finally, extensions to the K-coupon collector problem and the birthday problem are given.
Epidemics on Random Graphs with Tunable Clustering
Authors: Britton Tom , Deijfen Maria , Lagerås Andreas N. , Lindholm Mathias
Published: July 2016

In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.
The Number of Two Consecutive Successes in a Hoppe-Pólya Urn
Authors: Holst Lars
Published: July 2016

In a sequence of independent Bernoulli trials the probability of success in the kth trial is p

k
= a / (a + b + k − 1). An explicit formula for the binomial moments of the number of two consecutive successes in the first n trials is obtained and some consequences of it are derived.
How Nearly do Arriving Customers See Time-Average Behavior?
Authors: Peköz Erol A. , Ross Sheldon M. , Seshadri Sridhar
Published: July 2016

Customers arriving at a queue do not usually see its time-average behavior unless arrivals occur according to a Poisson process. In this article we study how nearly customers see time-average behavior. We give total variation error bounds for comparing the distance between the time- and customer-average distributions of a queueing system in terms of properties of the interarrival distribution. Some refinements are given for special cases and numerical computations are used to demonstrate the performance of the inequalities.
Entry and Exit Decision Problem with Implementation Delay
Authors: Costeniuc Marius , Schnetzer Michaela , Taschini Luca
Published: July 2016

We study investment and disinvestment decisions in situations where there is a time lagd> 0 from the timetwhen the decision is taken to the timet+dwhen the decision is implemented. In this paper we apply the probabilistic approach to the combined entry and exit decisions under the Parisian implementation delay. In particular, we prove the independence between Parisian stopping times and a general Brownian motion with drift stopped at the stopping time. Relying on this result, we solve the constrained maximization problem, obtaining an analytic solution to the optimal ‘starting’ and ‘stopping’ levels. We compare our results with the instantaneous entry and exit situation, and show that an increase in the uncertainty of the underlying process hastens the decision to invest or disinvest, extending a result of Bar-Ilan and Strange (1996).
A Random Permutation Model Arising in Chemistry
Authors: Brown Mark , Peköz Erol A. , Ross Sheldon M.
Published: July 2016

We study a model arising in chemistry where n elements numbered 1, 2, …, n are randomly permuted and if i is immediately to the left of i + 1 then they become stuck together to form a cluster. The resulting clusters are then numbered and considered as elements, and this process keeps repeating until only a single cluster is remaining. In this article we study properties of the distribution of the number of permutations required.
A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes
Authors: Holst Lars
Published: July 2016

A sequence of independent Bernoulli random variables with success probabilities a / (a + b + k − 1), k = 1, 2, 3, …, is embedded in a marked Poisson process with intensity 1. Using this, conditional Poisson limits follow for counts of failure strings.
A Continuous-Time Approach to Robbins' Problem of Minimizing the Expected Rank
Authors: Bruss F. Thomas , Swan Yvik C.
Published: July 2016

Let X
1, X
2, …, X
n be independent random variables uniformly distributed on [0,1]. We observe these sequentially and have to stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation? What is the value of the expected rank (as a function of n) and what is the limit of this value when n goes to ∞? This full-information expected selected-rank problem is known as Robbins' problem of minimizing the expected rank, and its general solution is unknown. In this paper we provide an alternative approach to Robbins' problem. Our model is similar to that of Gnedin (2007). For this, we consider a continuous-time version of the problem in which the observations follow a Poisson arrival process on ℝ+ × [0,1] of homogeneous rate 1. Translating the previous optimal selection problem in this setting, we prove that, under reasonable assumptions, the corresponding value function w
(t) is bounded and Lipschitz continuous. Our main result is that the limiting value of the Poisson embedded problem exists and is equal to that of Robbins' problem. We prove that w
(t) is differentiable and also derive a differential equation for this function. Although we have not succeeded in using this equation to improve on bounds on the optimal limiting value, we argue that it has this potential.
Exponential Random Graphs as Models of Overlay Networks
Authors: Draief M. , Ganesh A. , Massoulié L.
Published: July 2016

In this paper we give an analytic solution for graphs with n nodes and E = cn log n edges for which the probability of obtaining a given graph G is µn
(G) = exp (- β ∑i=1
n
d
i
2), where d
i is the degree of node i. We describe how this model appears in the context of load balancing in communication networks, namely peer-to-peer overlays. We then analyse the degree distribution of such graphs and show that the degrees are concentrated around their mean value. Finally, we derive asymptotic results for the number of edges crossing a graph cut and use these results (i) to compute the graph expansion and conductance, and (ii) to analyse the graph resilience to random failures.
Perturbed MAP Risk Models with Dividend Barrier Strategies
Authors: Cheung Eric C. K. , Landriault David
Published: July 2016

In the context of a dividend barrier strategy (see, e.g. Lin, Willmot and Drekic (2003)) we analyze the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process (MAP). We show that a relationship similar to the dividend-penalty identity of Gerber, Lin and Yang (2006) can be established for the class of perturbed MAP surplus processes, extending in the process some results of Li and Lu (2008) for the Markov-modulated risk model. Also, we revisit the same ruin-related quantities in an identical MAP risk model with the only exception that the barrier level effective at time t depends on the state of the underlying environment at this time. Similar relationships are investigated and derived. Numerical examples are also considered.