bluuebunny/crossref_metadata_2025 huggingface dataset:
A monotonicity result for a G/GI/c queue with balking or reneging
Authors: Ziya Serhan , Ayhan Hayriye , Foley Robert D. , Peköz Erol
Published: July 2016

In a G/GI/c loss system with balking, reneging, or limited waiting space, deleting some of the arriving customers can either increase or decrease the fraction of the remaining arrivals who get served, depending on how customers are deleted. We present a model in which the random deletion of arrivals independently and with some fixed probability can never decrease the fraction of the remaining arrivals who get served.
Mean Residual Lifetimes of Consecutive-k-out-of-n Systems
Authors: Navarro Jorge , Eryilmaz Serkan
Published: July 2016

In this paper we study reliability properties of consecutive-k-out-of-n systems with exchangeable components. For 2k ≦ n, we show that the reliability functions of these systems can be written as negative mixtures (i.e. mixtures with some negative weights) of two series (or parallel) systems. Some monotonicity and asymptotic properties for the mean residual lifetime function are obtained and some ordering properties between these systems are established. We prove that, under some assumptions, the mean residual lifetime function of the consecutive-k-out-of-n: G system (i.e. a system that functions if and only if at least k consecutive components function) is asymptotically equivalent to that of a series system with k components. When the components are independent and identically distributed, we show that consecutive-k-out-of-n systems are ordered in the likelihood ratio order and, hence, in the mean residual lifetime order, for 2k ≦ n. However, we show that this is not necessarily true when the components are dependent.
On Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Limited Playing Times
Authors: Yao Yi-Ching
Published: July 2016

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 - r)/r times the stake with probability w and losing the stake with probability 1 - w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes. This problem has been extended in several directions to account for limited playing time or future discounting. We propose a unifying framework that covers these extensions, and prove that bold play is optimal provided that w ≤ ½ ≤ r. We also show that this condition is in fact necessary for bold play to be optimal subject to the constraint of limited playing time.
Probabilistic Analysis of Unit-Demand Vehicle Routeing Problems
Authors: Bompadre Agustín , Dror Moshe , Orlin James B.
Published: July 2016

We analyze the unit-demand Euclidean vehicle routeing problem, where n customers are modeled as independent, identically distributed uniform points and have unit demand. We show new lower bounds on the optimal cost for the metric vehicle routeing problem and analyze them in this setting. We prove that there exists a constant ĉ > 0 such that the iterated tour partitioning heuristic given by Haimovich and Rinnooy Kan (1985) is a (2 - ĉ)-approximation algorithm with probability arbitrarily close to 1 as the number of customers goes to ∞. It has been a longstanding open problem as to whether one can improve upon the factor of 2 given by Haimovich and Rinnooy Kan. We also generalize this, and previous results, to the multidepot case.
Some Contributions to the Theory of Near-Critical Bisexual Branching Processes
Authors: Molina M. , Mota M. , Ramos A.
Published: July 2016

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.
Nonconstant Sum Red-And-Black Games with Bet-Dependent Win Probability Function
Authors: Pontiggia Laura
Published: July 2016

In this paper we investigate a class of N-person nonconstant sum red-and-black games with bet-dependent win probability functions. We assume that N players and a gambling house are engaged in a game played in stages, where the player's probability of winning at each stage is a function f of the ratio of his bet to the sum of all the players' bets. However, at each stage of the game there is a positive probability that all the players lose and the gambling house wins their bets. We prove that if the win probability function is super-additive and it satisfies f(s)f(t)≤f(st), then a bold strategy is optimal for all players.
Strong Convergence for URN Models with Reducible Replacement Policy
Authors: Abraham R. , Dhersin J. S. , Ycart B.
Published: July 2016

A multitype urn scheme with random replacements is considered. Each time a ball is picked, another ball is added, and its type is chosen according to the transition probabilities of a reducible Markov chain. The vector of frequencies is shown to converge almost surely to a random element of the set of stationary measures of the Markov chain. Its probability distribution is characterized as the solution to a fixed point problem. It is proved to be Dirichlet in the particular case of a single transient state to which no return is possible. This is no longer the case, however, as soon as returns to transient states are allowed.
The Critical Galton-Watson Process Without Further Power Moments
Authors: Nagaev S. V. , Wachtel V.
Published: July 2016

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z

n
; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.
The Pareto Copula, Aggregation of Risks, and the Emperor's Socks
Authors: Klüppelberg Claudia , Resnick Sidney I.
Published: July 2016

The copula of a multivariate distribution is the distribution transformed so that one-dimensional marginal distributions are uniform. We review a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions, and we call the resulting distribution the Pareto copula. Use of the Pareto copula has a certain claim to naturalness when considering asymptotic limit distributions for sums, maxima, and empirical processes. We discuss implications for aggregation of risk and offer some examples.
Beat the Mean: Sequential Selection by Better Than Average Rules
Authors: Krieger Abba M. , Pollak Moshe , Samuel-Cahn Ester
Published: July 2016

We consider a sequential rule, where an item is chosen into the group, such as a university faculty member, only if his/her score is better than the average score of those already belonging to the group. We study four variables: the average score of the members of the group after k items have been selected, the time it takes (in terms of the number of observed items) to assemble a group of k items, the average score of the group after n items have been observed, and the number of items kept after the first n items have been observed. We develop the relationships between these variables, and obtain their asymptotic behavior as k (respectively, n) tends to ∞. The assumption throughout is that the items are independent and identically distributed with a continuous distribution. Though knowledge of this distribution is not needed to implement the selection rule, the asymptotic behavior does depend on the distribution. We study in some detail the exponential, Pareto, and beta distributions. Generalizations of the ‘better than average’ rule to the β better than average rules are also considered. These are rules where an item is admitted to the group only if its score is better than β times the present average of the group, where β > 0.
A Collector's Problem with Renewal Arrival Processes
Authors: Kella Offer , Stadje Wolfgang
Published: July 2016

We study a collector's problem with K renewal arrival processes for different type items, where the objective is to collect complete sets. In particular, we derive the asymptotic distribution of the sequence of interarrival times between set completions.
Hitting Time and Inverse Problems for Markov Chains
Authors: de la Peña Victor , Gzyl Henryk , McDonald Patrick
Published: July 2016

Let W

n
be a simple Markov chain on the integers. Suppose that X

n
is a simple Markov chain on the integers whose transition probabilities coincide with those of W

n
off a finite set. We prove that there is an M > 0 such that the Markov chain W

n
and the joint distributions of the first hitting time and first hitting place of X

n
started at the origin for the sets {-M, M} and {-(M + 1), (M + 1)} algorithmically determine the transition probabilities of X

n
.
Asymptotics of Posteriors for Binary Branching Processes
Authors: Piau Didier
Published: July 2016

We compute the posterior distributions of the initial population and parameter of binary branching processes in the limit of a large number of generations. We compare this Bayesian procedure with a more naïve one, based on hitting times of some random walks. In both cases, central limit theorems are available, with explicit variances.
First Passage Times for Markov Additive Processes with Positive Jumps of Phase Type
Authors: Breuer Lothar
Published: July 2016

The present paper generalises some results for spectrally negative Lévy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration to compute this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.
Some Blackwell-Type Renewal Theorems for Weighted Renewal Functions
Authors: Lin Jianxi
Published: July 2016

In this paper, a new approach is proposed to investigate Blackwell-type renewal theorems for weighted renewal functions systematically according to which of the tails of weighted renewal constants or the underlying distribution is asymptotically heavier. Some classical results are improved considerably.
A Fourth Moment Inequality for Functionals of Stationary Processes
Authors: Durieu Olivier
Published: July 2016

In this paper, a fourth moment bound for partial sums of functionals of strongly ergodic Markov chains is established. This type of inequality plays an important role in the study of the empirical process invariance principle. This inequality is specially adapted to the technique of Dehling, Durieu, and Volný (2008). The same moment bound can be proved for dynamical systems whose transfer operator has some spectral properties. Examples of applications are given.
Chaoticity for Multiclass Systems and Exchangeability Within Classes
Authors: Graham Carl
Published: July 2016

Classical results for exchangeable systems of random variables are extended to multiclass systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multiclass system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within {each} class, and that a family of such systems converges in law {if and only if} the corresponding empirical measure vectors converge in law. As a corollary, convergence within {each} class to an infinite independent and identically distributed system implies asymptotic independence between {different} classes. A result implying the Hewitt-Savage 0-1 law is also extended.
Asymptotic Properties of a Mean-Field Model with a Continuous-State-Dependent Switching Process
Authors: Xi Fubao , Yin G.
Published: July 2016

This work is concerned with a class of mean-field models given by a switching diffusion with a continuous-state-dependent switching process. Focusing on asymptotic properties, the regularity or nonexplosiveness, Feller continuity, and strong Feller continuity are established by means of introducing certain auxiliary processes and by making use of the truncations. Based on these results, exponential ergodicity is obtained under the Foster–Lyapunov drift conditions. By virtue of the coupling methods, the strong ergodicity or uniform ergodicity in the sense of convergence in the variation norm is established for the mean-field model with a Markovian switching process. Besides this, several examples are presented for demonstration and illustration.
Bounds for the Distance Between the Distributions of Sums of Absolutely Continuous i.i.d. Convex-Ordered Random Variables with Applications
Authors: Christofides Tasos C. , Vaggelatou Eutichia
Published: July 2016

Let X
1, X
2,… and Y
1, Y
2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(X
i)=E(Y
i), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1
n
X
i and ∑i=1
n
Y
i. In the case where the distributions of the X
is and the Y
is are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.
Necklace Processes Via Pólya Urns
Authors: Nakata Toshio
Published: July 2016

Mallows and Shepp (2008) developed the following necklace processes. Start with a necklace consisting of one white bead and one black bead, and insert, one at a time, under a deterministic rule, a white bead or a black bead between a randomly chosen adjacent pair. They studied the statistical properties of the number of white beads by investigating the nature of the moments and the expected number of gaps of given length between white beads. In this note we study the number of white beads via Pólya urns and give a classification of necklace processes for some general rules. Additionally, we discuss the number of runs, i.e. the number of consecutive same color beads, instead of the number of gaps.
Extinction Times in Multitype Markov Branching Processes
Authors: Heinzmann Dominik
Published: July 2016

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.
Optimal test Interval for a Monotone Safety System
Authors: Aven Terje
Published: July 2016

We consider a safety system represented by a monotone (coherent) structure function of n components. The state of the components and the system is only revealed through inspection, which is carried out at intervals of length T. If the inspection shows that the system is in a critical state or has failed, it is overhauled and all components are restored to a good-as-new condition. Costs are associated with tests, system downtime, and repairs. The problem is to find an optimal T minimizing the expected long-run cost per unit of time. The purpose of this paper is to present a formal set-up for this problem and to show how an optimal T can be determined. A special case where the components have three states is given particular attention. It corresponds to a ‘delay time type system’, where the presence of a fault in a component does not lead to an immediate failure—there will be a ‘delay time’ between the occurrence of the fault and the failure of the component.
Comparisons of Parallel Systems According to the Convex Transform Order
Authors: Kochar Subhash , Xu Maochao
Published: July 2016

A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and homogeneous exponential samples in the sense of the dispersive order and the right-spread order are established. A sufficient condition is also given for the proportional hazard rate model.
Maximizing the Expected Duration of Owning a Relatively Best Object in a Poisson Process with Rankable Observations
Authors: Kurushima Aiko , Ano Katsunori
Published: July 2016

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity λ on some fixed time interval [0,T]. We assume a gamma prior density G
λ(r, 1/a) for λ. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time s

i

(r) onwards. The value of s

i

(r) can be obtained for each r and i as the unique root of a deterministic equation.
Quasi-Birth-and-Death Processes, Lattice Path Counting, and Hypergeometric Functions
Authors: van Leeuwaarden Johan S. H. , Squillante Mark S. , Winands Erik M. M.
Published: July 2016

In this paper we consider a class of quasi-birth-and-death processes for which explicit solutions can be obtained for the rate matrix
R
and the associated matrix
G
. The probabilistic interpretations of these matrices allow us to describe their elements in terms of paths on the two-dimensional lattice. Then determining explicit expressions for the matrices becomes equivalent to solving a lattice path counting problem, the solution of which is derived using path decomposition, Bernoulli excursions, and hypergeometric functions. A few applications are provided, including classical models for which we obtain some new results.
Poisson Approximation in a Poisson Limit Theorem Inspired by Coupon Collecting
Authors: Pósfai Anna
Published: July 2016

In this paper we refine a Poisson limit theorem of Gnedenko and Kolmogorov (1954): we determine the error order of a Poisson approximation for sums of asymptotically negligible integer-valued random variables that converge in distribution to the Poisson law. As an application of our results, we investigate the case of the coupon collector's problem when the distribution of the collector's waiting time is asymptotically Poisson.
Asymptotics of Implied Volatility far from Maturity
Authors: Tehranchi Michael R.
Published: July 2016

This note explores the behaviour of the implied volatility of a European call option far from maturity. Asymptotic formulae are derived with precise control over the error terms. The connection between the asymptotic implied volatility and the cumulant generating function of the logarithm of the underlying stock price is discussed in detail and illustrated by examples.
Moderate Deviations for Word Counts in Biological Sequences
Authors: Behrens Sarah , Löwe Matthias
Published: July 2016

We derive a moderate deviation principle for word counts (which is extended to counts of multiple patterns) in biological sequences under different models: independent and identically distributed letters, homogeneous Markov chains of order 1 and m, and, in view of the codon structure of DNA sequences, Markov chains with three different transition matrices. This enables us to approximate P-values for the number of word occurrences in DNA and protein sequences in a new manner.
The Busy Period of an M/G/1 Queue with Customer Impatience
Authors: Boxma Onno , Perry David , Stadje Wolfgang , Zacks Shelley
Published: July 2016

We consider an M/G/1 queue in which an arriving customer does not enter the system whenever its virtual waiting time, i.e. the amount of work seen upon arrival, is larger than a certain random patience time. We determine the busy period distribution for various choices of the patience time distribution. The main cases under consideration are exponential patience and a discrete patience distribution.
Inverse Stochastic Dominance, Majorization, and Mean Order Statistics
Authors: De La Cal Jesús , Cárcamo Javier
Published: July 2016

The inverse stochastic dominance of degree r is a stochastic order of interest in several branches of economics. We discuss it in depth, the central point being the characterization in terms of the weak r-majorization of the vectors of expected order statistics. The weak r-majorization (a notion introduced in the paper) is a natural extension of the classical (reverse) weak majorization of Hardy, Littlewood and Pòlya. This work also shows the equivalence between the continuous majorization (of higher order) and the discrete r-majorization. In particular, our results make it clear that the cases r = 1, 2 differ substantially from those with r ≥ 3, a fact observed earlier by Muliere and Scarsini (1989), among other authors. Motivated by this fact, we introduce new stochastic orderings, as well as new social inequality indices to compare the distribution of the wealth in two populations, which could be considered as natural extensions of the first two dominance rules and the S-Gini indices, respectively.
Efficient importance sampling in ruin problems for multidimensional regularly varying random walks
Authors: Blanchet Jose , Liu Jingchen
Published: July 2016

We consider the problem of efficient estimation via simulation of first passage time probabilities for a multidimensional random walk with heavy-tailed increments. In addition to being a natural generalization to the problem of computing ruin probabilities in insurance - in which the focus is the maximum of a one-dimensional random walk with negative drift - this problem captures important features of large deviations for multidimensional heavy-tailed processes (such as the role played by the mean of the process in connection to the location of the target set). We develop a state-dependent importance sampling estimator for this class of multidimensional problems. Then, using techniques based on Lyapunov inequalities, we argue that our estimator is strongly efficient in the sense that the relative mean squared error of our estimator can be made arbitrarily small by increasing the number of replications, uniformly as the probability of interest approaches 0.
On reliability prediction and semi-renewal processes
Authors: Tortorella Michael
Published: July 2016

We analyze the notion of ‘reliability prediction’ by studying in detail a key property that is tacitly assumed to make reliability prediction possible. The analysis leads in turn to a special type of point process for which the connection of future to past can be explicitly displayed. In this type of process, the semi-renewal process, all finite-dimensional distributions are completely determined by the distribution of the time to the first event in the process. The theory provides a heretofore unappreciated unification of the two most commonly used reliability prediction models for maintained systems, namely, the renewal and revival processes. We show that familiar results from renewal theory extend and generalize to semi-renewal processes.
Limit Theorems for a Generalized ST Petersburg Game
Authors: Gut Allan
Published: July 2016

The topic of the present paper is a generalized St Petersburg game in which the distribution of the payoff X is given by P(X =
sr
(k-1)/α) = pq

k-1,
k = 1, 2,…, where p + q = 1, s = 1 / p,
r = 1 / q, and 0 < α ≤ 1. For the case in which α = 1, we extend Feller's classical weak law and Martin-Löf's theorem on convergence in distribution along the 2
n
-subsequence. The analog for 0 < α < 1 turns out to converge in distribution to an asymmetric stable law with index α. Finally, some limit theorems for polynomial and geometric size total gains, as well as for extremes, are given.
An Almost-Sure Renewal Theorem for Branching Random Walks on the Line
Authors: Meiners Matthias
Published: July 2016

In the present paper an almost-sure renewal theorem for branching random walks (BRWs) on the real line is formulated and established. The theorem constitutes a generalization of Nerman's theorem on the almost-sure convergence of Malthus normed supercritical Crump-Mode-Jagers branching processes counted with general characteristic and Gatouras' almost-sure renewal theorem for BRWs on a lattice.
Poisson Approximation of the Number of Cliques in Random Intersection Graphs
Authors: Rybarczyk Katarzyna , Stark Dudley
Published: July 2016

A random intersection graphG(n,m,p) is defined on a setVofnvertices. There is an auxiliary setWconsisting ofmobjects, and each vertexv∈Vis assigned a random subset of objectsWv⊆Wsuch thatw∈Wvwith probabilityp, independently for allv∈Vand allw∈W. Given two verticesv1,v2∈V, we setv1∼v2if and only ifWv1∩Wv2≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number ofh-cliques inG(n,m,p) and a related Poisson distribution for any fixed integerh.
Positive Dependence of Signals
Authors: Denuit Michel
Published: July 2016

In this paper we further investigate the problem considered by Mizuno (2006) in the special case of identically distributed signals. Specifically, we first propose an alternative sufficient condition of crossing type for the convex order to hold between the conditional expectations given signal. Then, we prove that the bivariate (2,1)-increasing convex order ensures that the conditional expectations are ordered in the convex sense. Finally, the L
2 distance between the quantity of interest and its conditional expectation given signal (or expected conditional variance) is shown to decrease when the strength of the dependence increases (as measured by the (2,1)-increasing convex order).
Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions
Authors: Kuznetsov A.
Published: July 2016

In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.
Markov-Modulated Brownian Motion with Two Reflecting Barriers
Authors: Ivanovs Jevgenijs
Published: July 2016

We consider a Markov-modulated Brownian motion reflected to stay in a strip [0, B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explain its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent, exponentially distributed epoch. Our second contribution concerns transient behavior of the model. We identify the joint law of the processes defining the model at inverse local times.
A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk
Authors: Allaart Pieter
Published: July 2016

Let (B

t
)0≤t≤T
be either a Bernoulli random walk or a Brownian motion with drift, and let M

t
:= max{B
s: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤T
E[f(M

T
− B
τ], where the supremum is over all stopping times τ adapted to the natural filtration of (B

t
) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (B

t
) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.
Assessing the Reliability Function of Nanocomponents
Authors: Ebrahimi Nader
Published: July 2016

A nanocomponent is a collection of atoms arranged to a specific design in order to achieve a desired function with an acceptable performance and reliability. The type of atoms, the manner in which they are arranged within the nanocomponent, and their interrelationship have a direct effect on the nanocomponent's reliability (survival) function. In this paper we propose models based on the notion of a copula that are used to describe the relationship between the atoms of a nanocomponent. Having defined these models, we go on to construct a ‘nanocomponent’ model in order to obtain the reliability function of a nanocomponent.
Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes
Authors: Löpker Andreas , Stadje Wolfgang
Published: July 2016

We consider the level hitting times τy= inf{t≥ 0 |Xt=y} and the running maximum processMt= sup{Xs| 0 ≤s≤t} of a growth-collapse process (Xt)t≥0, defined as a [0, ∞)-valued Markov process that grows linearly between random ‘collapse’ times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τycan be determined in terms of the extended generator ofXtand give a power series expansion of the reciprocal of Ee−sτy. We prove asymptotic results for τyandMt: for example, ifm(y) = Eτyis of rapid variation thenMt/m-1(t) →w1 ast→ ∞, wherem-1is the inverse function ofm, while ifm(y) is of regular variation with indexa∈ (0, ∞) andXtis ergodic, thenMt/m-1(t) converges weakly to a Fréchet distribution with exponenta. In several special cases we provide explicit formulae.
A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process
Authors: Thomas Peter J.
Published: July 2016

We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck processX(t) obeying dX= -βXdt+ σdWto reach a fixed threshold θ from a suprathreshold initial conditionx0> θ > 0 has a lower bound of the form ρ(t) >kexp[-pe6βt] for positive constantskandpfor timestexceeding some positive valueu. We obtain explicit expressions fork,p, anduin terms of β, σ,x0, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.
A Central Limit Theorem and its Applications to Multicolor Randomly Reinforced Urns
Authors: Berti Patrizia , Crimaldi Irene , Pratelli Luca , Rigo Pietro
Published: July 2016

Let X

n
be a sequence of integrable real random variables, adapted to a filtration (G

n
). Define C

n
= √{(1 / n)∑
k=1

n

X

k
− E(X

n+1 | G

n
)} and D

n
= √n{E(X

n+1 | G

n
) − Z}, where Z is the almost-sure limit of E(X

n+1 | G

n
) (assumed to exist). Conditions for (C

n
, D

n
) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑
k=1

n
X_k - Z} = C

n
+ D

n
→ N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.
The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks
Authors: Chen Yiqing
Published: July 2016

Consider a discrete-time insurance risk model. Within periodi, the net insurance loss is denoted by a real-valued random variableXi. The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factorYifrom timeito timei− 1. Assume that (Xi,Yi),i∈N, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functionsFandG. WhenFis subexponential andGfulfills some constraints in order for the product convolution ofFandGto be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in whichFbelongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.
A class of complete benchmark models with intensity-based jumps
Authors: Platen Eckhard
Published: July 2016

This paper proposes a class of complete financial market models, the benchmark models, with security price processes that exhibit intensity-based jumps. The benchmark or reference unit is chosen to be the growth-optimal portfolio. Primary security account prices, when expressed in units of the benchmark, turn out to be local martingales. In the proposed framework an equivalent risk-neutral measure need not exist. Benchmarked fair derivative prices are obtained as conditional expectations of future benchmarked prices under the real-world probability measure. This concept of fair pricing generalizes the classical risk-neutral approach and the actuarial present-value pricing methodology.
Precise large deviations for sums of random variables with consistently varying tails
Authors: Ng Kai W. , Tang Qihe , Yan Jia-An , Yang Hailiang
Published: July 2016

Let {X

k
, k ≥ 1} be a sequence of independent, identically distributed nonnegative random variables with common distribution function F and finite expectation μ > 0. Under the assumption that the tail probability is consistently varying as x tends to infinity, this paper investigates precise large deviations for both the partial sums S

n
and the random sums S

N(t), where N(·) is a counting process independent of the sequence {X

k
, k ≥ 1}. The obtained results improve some related classical ones. Applications to a risk model with negatively associated claim occurrences and to a risk model with a doubly stochastic arrival process (extended Cox process) are proposed.
A note on two-level superprocesses
Authors: Hong Wen-Ming
Published: July 2016

We prove some central limit theorems for a two-level super-Brownian motion with random immigration, which lead to limiting Gaussian random fields. The covariances of those Gaussian fields are explicitly characterized.
Russian options with a finite time horizon
Authors: Ekström Erik
Published: July 2016

We investigate the Russian option with a finite time horizon in the standard Black–Scholes model. The value of the option is shown to be a solution of a certain parabolic free boundary problem, and the optimal stopping boundary is shown to be continuous. Moreover, the asymptotic behavior of the optimal stopping boundary near expiration is studied.
Convergence to the coalescent in populations of substantially varying size
Authors: Jagers Peter , Sagitov Serik
Published: July 2016

Kingman's classical theory of the coalescent uncovered the basic pattern of genealogical trees of random samples of individuals in large but time-constant populations. Time is viewed as being discrete and is identified with non-overlapping generations. Reproduction can be very generally taken as exchangeable (meaning that the labelling of individuals in each generation carries no significance). Recent generalisations have dealt with population sizes exhibiting given deterministic or (minor) random fluctuations. We consider population sizes which constitute a stationary Markov chain, explicitly allowing large fluctuations in short times. Convergence of the genealogical tree, as population size tends to infinity, towards the (time-scaled) coalescent is proved under minimal conditions. As a result, we obtain a formula for effective population size, generalising the well-known harmonic mean expression for effective size.
Comparison of multivariate risks and positive dependence
Authors: Rüschendorf Ludger
Published: July 2016

In this paper we extend some recent results on the comparison of multivariate risk vectors with respect to supermodular and related orderings. We introduce a dependence notion called the ‘weakly conditional increasing in sequence order’ that allows us to conclude that ‘more dependent’ vectors in this ordering are also comparable with respect to the supermodular ordering. At the same time, this ordering allows us to compare two risks with respect to the directionally convex order if the marginals increase convexly. We further state comparison criteria with respect to the directionally convex order for some classes of risk vectors which are modelled by functional influence factors. Finally, we discuss Fréchet bounds with respect to Δ-monotone functions when multivariate marginals are given. It turns out that, in the case of multivariate marginals, comonotone vectors no longer yield necessarily the largest risks but, in some cases, may even be vectors which minimize risk.
Limit theorems for iterated random functions
Authors: Wu Wei Biao , Shao Xiaofeng
Published: July 2016

We study geometric moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under a Dini-continuity condition, a central limit theorem for additive functionals of such systems is established. The empirical processes of sample paths are shown to converge to Gaussian processes in the Skorokhod space. An exponential inequality is established. We present a bound for joint cumulants, which ensures the applicability of several asymptotic results in spectral analysis of time series. Our results provide a vehicle for statistical inferences for fractals and many nonlinear time series models.
Optimal stopping rules for correlated random walks with a discount
Authors: Allaart Pieter
Published: July 2016

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.
A stable algorithm for stationary distribution calculation for a BMAP/SM/1 queueing system with Markovian arrival input of disasters
Authors: Dudin Alexander , Semenova Olga
Published: July 2016

Disaster arrival into a queueing system causes all customers to leave the system instantaneously. We present a numerically stable algorithm for calculating the stationary state distribution of an embedded Markov chain for the BMAP/SM/1 queue with a MAP input of disasters.
Infinite- and finite-buffer Markov fluid queues: a unified analysis
Authors: Akar Nail , Sohraby Khosrow
Published: July 2016

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.
On Markov-dependent parking problems
Authors: Yang Jiing-Ru , Hsiau Shoou-Ren
Published: July 2016

We drive a car along a street towards our destination and look for an available parking place without turning around. Each parking place is associated with a loss which decreases with the distance of the parking place from our destination. Assume that the states (empty or filled) of the parking places form a Markov chain. We want to find an optimal parking strategy to minimize the expected loss. A curious example is constructed and two sufficient conditions for the existence of the threshold-type optimal parking strategy are given.
Stationary Time Series Models with Exponential Dispersion Model Margins
Authors: Jørgensen Bent , Xue-Kun Song Peter
Published: July 2016

We consider a class of stationary infinite-order moving average processes with margins in the class of infinitely divisible exponential dispersion models. The processes are constructed by means of the thinning operation of Joe (1996), generalizing the binomial thinning used by McKenzie (1986, 1988) and Al-Osh and Alzaid (1987) for integer-valued time series. As a special case we obtain a class of autoregressive moving average processes that are different from the ARMA models proposed by Joe (1996). The range of possible marginal distributions for the new models is extensive and includes all infinitely divisible distributions with finite moment generating functions, hereunder many known discrete, continuous and mixed distributions.
Stochastic comparison of repairable systems by coupling
Authors: Last Günter , Szekli Ryszard
Published: July 2016

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.
A deterministic epidemic model taking account of repeated contacts between the same individuals
Authors: Diekmann O. , de Jong M. C. M. , Metz J. A. J.
Published: July 2016

We introduce a certain population contact structure and derive, in three different ways, the final size equation for a quite general superimposed epidemic process. The contact structure is characterized by the following two properties: (i) each individual contacts exactly k other individuals; (ii) these k acquaintances are a random sample of the (infinite) population.
Optimal and myopic search in a binary random vector
Authors: Dor Avner , Greenshtein Eitan , Korach Ephraim
Published: July 2016

Let X = (X
1, …, X

n
) be a random binary vector, with a known joint distribution P. It is necessary to inspect the coordinates sequentially in order to determine if X

i
= 0 for every i, i = 1, …, n. We find bounds for the ratio of the expected number of coordinates inspected using optimal and greedy searching policies.
On the behaviour of a long cascade of linear reservoirs
Authors: Glynn John E. , Glynn Peter W.
Published: July 2016

This paper describes the limiting asymptotic behaviour of a long cascade of linear reservoirs fed by stationary inflows into the first reservoir. We show that the storage in the nth reservoir becomes asymptotically deterministic as n → ∞, and establish a central limit theorem for the random fluctuations about the deterministic approximation. In addition, we prove a large deviations theorem that provides precise logarithmic asymptotics for the tail probabilities associated with the storage in the nth reservoir when n is large.
The correlated random walk with boundaries: A combinatorial solution
Authors: Böhm W.
Published: July 2016

The transition functions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes.
Estimation of limiting availability for a stationary bivariate process
Authors: Abraham B. , Balakrishna N.
Published: July 2016

We estimate the limiting availability of a system when the operating and repair times form a stationary bivariate sequence. These estimators are shown to be consistent and asymptotically normal under certain conditions. In particular, we estimate the limiting availability for a bivariate exponential autoregressive process.
Convergence to stationarity in the Moran model
Authors: Donnelly Peter , Rodrigues Eliane R.
Published: July 2016

Consider a population of fixed size consisting of N haploid individuals. Assume that this population evolves according to the two-allele neutral Moran model in mathematical genetics. Denote the two alleles by A
1 and A
2. Allow mutation from one type to another and let 0 < γ < 1 be the sum of mutation probabilities. All the information about the population is recorded by the Markov chain X = (X(t))
t≥0 which counts the number of individuals of type A
1. In this paper we study the time taken for the population to ‘reach’ stationarity (in the sense of separation and total variation distances) when initially all individuals are of one type. We show that after t
∗ = Nγ-1logN + cN the separation distance between the law of X(t
∗) and its stationary distribution converges to 1 - exp(-γe-γc
) as N → ∞. For the total variation distance an asymptotic upper bound is obtained. The results depend on a particular duality, and couplings, between X and a genealogical process known as the lines of descent process.
A note on the rate of convergence to equilibrium for Erlang's model in the subcritical case
Authors: Voit Michael
Published: July 2016

We derive some asymptotic results for the rate of convergence to equilibrium for the number of busy servers in an M/M/N/N queue with input rate λN and service rate 1 for N → ∞ in the ‘subcritical’ case λ ∈]0, 1[. These results improve recent contributions of Fricker, Robert and Tibi.
Statistical and information based (physical) minimal repair for k out of n Systems
Authors: Boland Philip J. , El-Neweihi Emad
Published: July 2016

The ‘minimal’ repair of a system can take several forms. Statistical or black box minimal repair at failure time t is equivalent to replacing the system with another functioning one of the same age, but without knowledge of precisely what went wrong with the system. Its major attribute is its mathematical tractability. In physical minimal repair, at system failure time t, we minimally repair the ‘component’ which brought the system down at time t. The work of Arjas and Norros, Finkelstein, and Natvig is reviewed. The concept of a rate function for minimal repairs of the statistical and physical types are discussed and developed. It is shown that the number of physical minimal repairs is stochastically larger than the number of statistical minimal repairs for k out of n systems with similar components. Some majorization results are given for physical minimal repair for two component parallel systems with exponential components.