bluuebunny/crossref_metadata_2025 huggingface dataset:
A series expansion approach to the inverse problem
Authors: Angulo J. M. , Ruiz-Medina M. D.
Published: July 2016

We consider the inverse problem of estimating the input random field in a stochastic integral equation relating two random fields. The purpose of this paper is to present an approach to this problem using a Riesz-based or orthonormal-based series expansion of the input random field with uncorrelated random coefficients. We establish conditions under which the input series expansion induces (via the integral equation) a Riesz-based or orthonormal-based series expansion for the output random field. The estimation problem is studied considering two cases, depending on whether data are available from either the output random field alone, or from both the input and output random fields. Finally, we discuss this approach in the case of transmissivity estimation from piezometric head data, which was the original motivation of this work.
Absorbing process in recursive stochastic equations
Authors: Nakatsuka Toshinao
Published: July 2016

We introduce the concept of the absorbing process for analysing a state process. Our aim is to show the existence of the absorbing process with probability one. This process is shown to be stationary, asymptotically stationary, periodic or a.m.s., if the input distribution has such properties. The real process is absorbed into this process so that its stability and some other properties are easily derived.
On identifiability in the autopsy model of reliability systems of lifemonitored and conditionally lifemonitored components
Authors: Gåsemyr Jørund
Published: July 2016

The autopsy data of a coherent system is the pair (T, I), where T is the time to the failure of the system, and I is the set of components failed by that time. A basic question is under which conditions the distribution of the autopsy data determines the joint life distribution of the components. This problem has been studied under various distributional and structural assumptions (always under the assumption of independent component life distributions) by Meilijson (1981), Nowik (1990) and Antoine, Doss and Hollander (1993). In this paper we study the same problem in the situation when the autopsy data is supplemented with data from lifemonitoring and conditional lifemonitoring of some components, as introduced by Meilijson (1994).
Expected measure of the union of random rectangles
Authors: Chao Chern-Ching , Hwang Hsien-Kuei , Liang Wen-Qi
Published: July 2016

An asymptotic expression for the expected area of the union of n random rectangles is derived by Mellin transforms, where their two diagonal corners are independently and uniformly distributed over (0,1)2. The general result for d-dimensional hyper-rectangles is also stated.
A comparison of the stationary distributions of GI/M/c/n and GI/M/c
Authors: Simonot F.
Published: July 2016

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/c/n and GI/M/c queueing systems as n tends to infinity. We show that earlier results established for GI/M/1/n and GI/M/1 remain true. Namely, it is proved that if the interarrival time c.d.f. H is non lattice with mean value λ−1 and if the traffic intensity is strictly less than one, then the convergence rates in l
1
norm of the arrival and time stationary distributions of GI/M/c/n to the corresponding stationary distributions of GI/M/c are geometric and are characterized by ω, the unique solution in (0,1) of the equation z = ∫∞
0 exp{-μc(1-z)t}dH(t).
The distributions of cluster functionals of extreme events in a dth-order Markov chain
Authors: Yun Seokhoon
Published: July 2016

The paper concerns the asymptotic distributions of cluster functionals of extreme events in a dth-order stationary Markov chain {X

n
, n = 1,2,…} for which the joint distribution of (X
1,…,X

d+1) is absolutely continuous. Under some distributional assumptions for {X

n
}, we establish weak convergence for a class of cluster functionals and obtain representations for the asymptotic distributions which are well suited for simulation. A number of examples important in applications are presented to demonstrate the usefulness of the results.
Optimal stopping with discount and observation costs
Authors: Kühne Robert , Rüschendorf Ludger
Published: July 2016

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.
Compound Poisson approximation for the Johnson-Mehl model
Authors: Erhardsson Torkel
Published: July 2016

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.
An exponential change of measure for the Feller diffusion and some implications for superprocesses
Authors: Adler Robert J. , Iyer Srikanth K.
Published: July 2016

Let X

t
be a Feller (branching) diffusion with drift αx. We consider new processes, the probability measures of which are obtained from that of X via changes of measure involving suitably normalized exponential functions of with λ > 0. The new processes can be thought of as ‘self-reinforcing’ versions of the old.

Depending on the values of α, T and λ, the process under the new measure is shown to exhibit explosion in finite time. We also obtain a number of other results related to the new processes.

Since the Feller diffusion is also the total mass process of a superprocess, we relate the finite-time explosion property to the behaviour of superprocesses with local self-interaction, and raise some interesting questions for these.
Passage time moments for multidimensional diffusions
Authors: Balaji S. , Ramasubramanian S.
Published: July 2016

Let τ
r
denote the hitting time of B(0:r) for a multidimensional diffusion process. We give verifiable criteria for finiteness/infiniteness of As an application we exhibit classes of diffusion processes which are recurrent but is infinite for all p > 0, |x| > r > 0; this includes the two-dimensional Brownian motion and the reflecting Brownian motion in a wedge with a certain parameter α = 0.
A note on bias optimality in controlled queueing systems
Authors: Lewis Mark E. , Puterman Martin L.
Published: July 2016

The use ofbias optimalityto distinguish among gain optimal policies was recently studied by Haviv and Puterman [1] and extended in Lewiset al.[2]. In [1], upon arrival to anM/M/1 queue, customers offer the gatekeeper a rewardR. If accepted, the gatekeeper immediately receives the reward, but is charged a holding cost,c(s), depending on the number of customers in the system. The gatekeeper, whose objective is to ‘maximize’ rewards, must decide whether to admit the customer. If the customer is accepted, the customer joins the queue and awaits service. Haviv and Puterman [1] showed there can be only two Markovian, stationary, deterministic gain optimal policies and that only the policy which uses thelargercontrol limit is bias optimal. This showed the usefulness of bias optimality to distinguish between gain optimal policies. In the same paper, they conjectured that if the gatekeeper receives the reward uponcompletionof a job instead of upon entry, the bias optimal policy will be the lower control limit. This note confirms that conjecture.
Probabilistic aspects of the SU(2)-harmonic oscillator
Authors: Luo Shunlong
Published: July 2016

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.
Explicit formulae for stationary distributions of stress release processes
Authors: Borovkov K. , Vere-Jones D.
Published: July 2016

Stress release processes are special Markov models attempting to describe the behaviour of stress and occurrence of earthquakes in seismic zones. The stress is built up linearly by tectonic forces and released spontaneously when earthquakes occur. Assuming that the risk is an exponential function of the stress, we derive closed form expressions for the stationary distribution of such processes, the moments of the risk, and the autocovariance function of the reciprocal risk process.
The Galton-Watson process revisited: some martingale relationships and applications
Authors: Lynch James D.
Published: July 2016

A martingale is used to study extinction probabilities of the Galton-Watson process using a stopping time argument. This same martingale defines a martingale function in its arguments; consequently, its derivative is also a martingale. The argumentscan be classified as regular or irregular and this classification dictates very different behavior of the Galton-Watson process. For example, it is shown that irregularity of a pointsis equivalent to the derivative martingale sequence atsbeing closable, (i.e., it has limit which, when attached to the original sequence, the martingale structure is retained). It is also shown that for irregular points the limit of the derivative is the derivative of the limit, and two different types of norming constants for the asymptotics of the Galton-Watson process are asymptotically equivalent only for irregular points.
Exponential convergence of adaptive importance sampling for Markov chains
Authors: Baggerly Keith , Cox Dennis , Picard Rick
Published: July 2016

We consider adaptive importance sampling for a Markov chain with scoring. It is shown that convergence to the zero-variance importance sampling chain for the mean total score occurs exponentially fast under general conditions. These results extend previous work in Kollman (1993) and in Kollman et al. (1999) for finite state spaces.
Stochastic models for dependent life lengths induced by common pure jump shock environments
Authors: Li Haijun
Published: July 2016

The lifelengths of components of a system are usually dependent due to the common random production and operating environments. In this paper, we introduce a multi-variate pure jump Markov process to describe a large class of damage processes on various system components driven by common environmental shocks, and establish some dependence properties (association) for such a process and its multivariate increment process. These strong association properties describe both spatial dependence and temporal dependence of a multivariate pure jump process, and also provide a vehicle to derive some structural properties of component lifelengths of the systems operating in such an environment. Some bounds for the joint survival functions of component lifelengths are also obtained.
An investment model with entry and exit decisions
Authors: Duckworth J. Kate , Zervos Mihail
Published: July 2016

We consider an investment model which generalizes a number of models that have been studied in the literature. The model involves entry and exit decisions as well as decisions relating to production scheduling. We then address the problem of its valuation from the standpoint of the dynamic programming approach. Our analysis results in a closed form analytic solution that can take qualitatively different forms depending on parameter values.
A probabilistic proof of non-explosion of a non-linear PDE system
Authors: López-Mimbela J. Alfredo , Wakolbinger Anton
Published: July 2016

Using a representation in terms of a two-type branching particle system, we prove that positive solutions of the system remain bounded for suitable bounded initial conditions, provided A and B generate processes with independent increments and one of the processes is transient with a uniform power decay of its semigroup. For the case of symmetric stable processes on R
1,this answers a question raised in [4].
Dynamics of large uncontrolled loss networks
Authors: Zachary Stan
Published: July 2016

This paper studies the connection between the dynamical and equilibrium behaviour of large uncontrolled loss networks. We consider the behaviour of the number of calls of each type in the network, and show that, under the limiting regime of Kelly (1986), all trajectories of the limiting dynamics converge to a single fixed point, which is necessarily that on which the limiting stationary distribution is concentrated. The approach uses Lyapunov techniques and involves the evolution of the transition rates of a stationary Markov process in such a way that it tends to reversibility.
A central limit theorem for iterated random functions
Authors: Wu Wei Biao , Woodroofe Michael
Published: July 2016

A central limit theorem is established for additive functions of a Markov chain that can be constructed as an iterated random function. The result goes beyond earlier work by relaxing the continuity conditions imposed on the additive function, and by relaxing moment conditions related to the random function. It is illustrated by an application to a Markov chain related to fractals.
Generalized reliability bounds for coherent structures
Authors: Boutsikas Michael V. , Koutras Markos V.
Published: July 2016

In this article we introduce generalizations of several well-known reliability bounds. These bounds are based on arbitrary partitions of the family of minimal path or cut sets of the system and can be used for approximating the reliability of any coherent structure with i.i.d. components. An illustration is also given of how the general results can be applied for a specific reliability structure (two-dimensional consecutive-k
1 x k
2-out-of-n
1x n
2 system) along with extensive numerical calculations revealing that, in most cases, the generalized bounds perform better than other available bounds in the literature for this system.
A stochastic lower bound for assemble-transfer batch service queueing networks
Authors: Economou Antonis
Published: July 2016

Miyazawa and Taylor (1997) introduced a class of assemble-transfer batch service queueing networks which do not have tractable stationary distribution. However by assuming a certain additional arrival process at each node when it is empty, they obtain a geometric product-form stationary distribution which is a stochastic upper bound for the stationary distribution of the original network. In this paper we develop a stochastic lower bound for the original network by introducing an additional departure process at each node which tends to remove all the customers present in it. This model in combination with the aforementioned upper bound model gives a better sense for the properties of the original network.
An insensitivity property of Lundberg's estimate for delayed claims
Authors: Brémaud Pierre
Published: July 2016

This short note shows that the Lundberg exponential upper bound in the ruin problem of non-life insurance with compound Poisson claims is also valid for the Poisson shot noise delayed-claims model, and that the optimal exponent depends only on the distribution of the total claim per accident, not on the time it takes to honour the claim. This result holds under Cramer's condition.
Dynamical systems defined on point processes
Authors: Baxter Laurence A.
Published: July 2016

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.
A central limit theorem for conditionally centred random fields with an application to Markov fields
Authors: Comets Francis , Janžura Martin
Published: July 2016

We prove a central limit theorem for conditionally centred random fields, under a moment condition and strict positivity of the empirical variance per observation. We use a random normalization, which fits non-stationary situations. The theorem applies directly to Markov random fields, including the cases of phase transition and lack of stationarity. One consequence is the asymptotic normality of the maximum pseudo-likelihood estimator for Markov fields in complete generality.
Partition structures and sufficient statistics
Authors: Joyce Paul
Published: July 2016

Is the Ewens distribution the only one-parameter family of partition structures where the total number of types sampled is a sufficient statistic? In general, the answer is no. It is shown that all counterexamples can be generated via an urn scheme. The urn scheme need only satisfy two general conditions. In fact, the conditions are both necessary and sufficient. However, in particular, for a large class of partition structures that naturally arise in the infinite alleles theory of population genetics, the Ewens distribution is the only one in this class where the total number of types is sufficient for estimating the mutation rate. Finally, asymptotic sufficiency for parametric families of partition structures is discussed.
On convergence and extensions of size-biased permutations
Authors: Gnedin Alexander V.
Published: July 2016

Size-biased permutation (SBP) is a random arrangement of frequencies of distinct categories in the order in which the categories appear for the first time in the sampling process. We study the conditions under which the SBPs converge in distribution and discuss extended versions of SBP for the case when the sum of positive frequencies is less than 1.
First passage percolation and a model for competing spatial growth
Authors: Häggström Olle , Pemantle Robin
Published: July 2016

An interacting particle system modelling competing growth on the ℤ2 lattice is defined as follows. Each x ∈ ℤ2 is in one of the states {0,1,2}. 1's and 2's remain in their states for ever, while a 0 flips to a 1 (a 2) at a rate equal to the number of its neighbours which are in state 1 (2). This is a generalization of the well-known Richardson model. 1's and 2's may be thought of as two types of infection, and 0's as uninfected sites. We prove that if we start with a single site in state 1 and a single site in state 2, then there is positive probability for the event that both types of infection reach infinitely many sites. This result implies that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.
A monotonicity result for the workload in Markov-modulated queues
Authors: Bäuerle Nicole , Rolski Tomasz
Published: July 2016

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.
Applying POS(i) rules to communication problems
Authors: Chong K. S. , Lam K.
Published: July 2016

A spectrum of self-organizing rules including the move-to-front rule and the transposition rule are applied to the communication problem. The stationary distributions under these rules are obtained. Cost comparison between them is considered. In the special case of three paths, it is shown that the transposition rule always outperforms the move-to-front rule.
A relatively quick way to simulate local random processes on a lattice
Authors: Richardson Daniel , Burton Wayne
Published: July 2016

A class of Markov processes in continuous time, with local transition rules, acting on colourings of a lattice, is defined. An algorithm is described for dynamic simulation of such processes. The computation time for the next state is O(logb), where b is the number of possible next states. This technique is used to give some evidence that the limiting shape for a random growth process in the plane with exponential distribution is approximately a circle.
Length of Galton–Watson trees and blow-up of semilinear systems
Authors: López-Mimbela J. Alfredo , Wakolbinger Anton
Published: July 2016

By lower estimates of the functionals 𝔼[e
S

t

K

t

N

t

], where S

t
and N

t
denote the total length up to time t and the number of individuals at time t in a Galton-Watson tree, we obtain sufficient criteria for the blow-up of semilinear equations and systems of the type ∂w

t
/∂t = A
w

t
+ V
w

t

β. Roughly speaking, the growth of the tree length has to win against the ‘mobility’ of the motion belonging to the generator A, since, in the probabilistic representation of the equations, the latter results in small K(t) as t → ∞. In the single-type situation, this gives a re-interpretation of classical results of Nagasawa and Sirao(1969); in the multitype scenario, part of the results obtained through analytic methods in Escobedo and Herrero (1991) and (1995) are re-proved and extended from the case A = Δ to the case of α-Laplacians.
Approximating integrals of stochastic processes: extensions
Authors: Benhenni Karim
Published: July 2016

We consider the problem of predicting integrals of second order processes whose covariances satisfy some Hölder regularity condition of order α > 0. When α is an odd integer, linear estimators based on regular sampling designs were constructed and asymptotic results for the approximation error were derived. We extend this result to any α > 0. When 2K < α ≤ 2K + 2, K a non-negative integer, we use an appropriate predictor based on the Euler-MacLaurin formula of order K with regular sampling designs. We give the corresponding result for the mean square error.
A uniform convergence theorem for the numerical solving of the nonlinear filtering problem
Authors: Del Moral P.
Published: July 2016

The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.
Simple conditions for the convergence of simulated annealing type algorithms
Authors: Cruz J. R. , Dorea C. C. Y.
Published: July 2016

We study a class of simulated annealing type algorithms for global minimization with general acceptance probabilities. This paper presents simple conditions, easy to verify in practice, which ensure the convergence of the algorithm to the global minimum with probability 1.
Weak convergence on randomly deleted sets
Authors: Rothmann Mark D. , Russo Ralph P.
Published: July 2016

Suppose t
1, t
2,… are the arrival times of units into a system. The kth entering unit, whose magnitude is X

k
and lifetime L

k
, is said to be ‘active’ at time t if I(t

k
< t

k
+ L

k
) = I

k,t
= 1. The size of the active population at time t is thus given by A

t
= ∑
k≥1
I

k,t
. Let V

t
denote the vector whose coordinates are the magnitudes of the active units at time t, in their order of appearance in the system. For n ≥ 1, suppose λ
n
is a measurable function on n-dimensional Euclidean space. Of interest is the weak limiting behaviour of the process λ*(t) whose value is λ
m
(V

t
) or 0, according to whether A

t
= m > 0 or A

t
= 0.
On some multi-request move-to-front heuristics
Authors: Phatarfod R. M. , Pryde A. J.
Published: July 2016

We consider the problem of self-organizing a linear list when the list is subjected to a random number of requests. Three separate move-to-front heuristics are considered. The difference between them lies in whether the items in the requested set are moved to the front in random order, or the same order as they were in originally, or in the order opposite to the one they were in originally. The eigenvalues of the transition probability matrices corresponding to the three heuristics are explicitly derived.
Queueing processes in bulk systems under the D-policy
Authors: Dshalalow Jewgeni H.
Published: July 2016

This paper studies the queueing process in a class of D-policy models with Poisson bulk input, general service time, and four different vacation scenarios, among them a multiple vacation, single vacation and idle server. The D-policy specifies a busy period discipline, which requires an idle or vacationing server to resume his service when the workload process crosses some fixed level D. The analysis of the queueing process is based on the theory of fluctuations for three-dimensional marked counting processes presented in the paper. For all models, we derive the stationary distributions for the embedded and continuous time parameter queueing processes in closed analytic forms and illustrate the results by a number of examples and applications.
Tight bound of random interval packing
Authors: Rhee Wansoo T.
Published: July 2016

Consider n random intervals I
1, …, I

N
chosen by selecting endpoints independently from the uniform distribution. A packing of I
1, …, I

N
is a disjoint sub-collection of these intervals: its wasted space is the measure of the set of points not covered by the packing. We investigate the random variable W

N
equal to the smallest wasted space among all packings. Coffman, Poonen and Winkler proved that EW

N
is of order (log N)2/N. We provide a sharp estimate of log P(W

N
≥ t (log N)2 / N) and log P(W

N
≤ t (log N)2 / N) for all values of t.
Simpson-type paradoxes, dependence, and ageing
Authors: Scarsini Marco , Spizzichino Fabio
Published: July 2016

We will state a general version of Simpson's paradox, which corresponds to the loss of some dependence properties under marginalization. We will then provide conditions under which the paradox is avoided. Finally we will relate these Simpson-type paradoxes to some well-known paradoxes concerning the loss of ageing properties when the level of information changes.
Extreme values of phase-type and mixed random variables with parallel-processing examples
Authors: Kang Sungyeol , Serfozo Richard F.
Published: July 2016

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.
On an inequality for the normal distribution arising in bioequivalence studies
Authors: Yao Yi-Ching , Iyer Hari
Published: July 2016

For (μ,σ2) ≠ (0,1), and 0 < z < ∞, we prove that

where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 < z < ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.
On spectral gap estimates of a Markov chain via hitting times and coupling
Authors: Meise Christian
Published: July 2016

Well-known inequalities for the spectral gap of a discrete-time Markov chain, such as Poincaré's and Cheeger's inequalities, do not perform well if the transition graph of the Markov chain is strongly connected. For example in the case of nearest-neighbour random walk on then-dimensional cube Poincaré's and Cheeger's inequalities are off by a factorn. Using a coupling technique and a contraction principle lower bounds on the spectral gap can be derived. Finally, we show that using the contraction principle yields a sharp estimate for nearest-neighbour random walk on then-dimensional cube.
The fundamental matrix for a certain random walk
Authors: Taylor Howard M.
Published: July 2016

Consider the random walk {Sn} whose summands have the distributionP(X=0) = 1-(2/π), andP(X= ±n) = 2/[π(4n2−1)], forn≥ 1. This random walk arises when a simple random walk in the integer plane is observed only at those instants at which the two coordinates are equal. We derive the fundamental matrix, or Green function, for the process on the integral [0,N] = {0,1,…,N}, and from this, an explicit formula for the mean timexkfor the random walk starting fromS0=kto exit the interval. The explicit formula yields the limiting behavior ofxkasN→ ∞ withkfixed. For the random walk starting from zero, the probability of exiting the interval on the right is obtained. By lettingN→ ∞ in the fundamental matrix, the Green function on the interval [0,∞) is found, and a simple and explicit formula for the probability distribution of the point of entry into the interval (−∞,0) for the random walk starting fromk= 0 results. The distributions for some related random variables are also discovered.

Applications to stress concentration calculations in discrete lattices are briefly reviewed.
Central limit theorems for multivariate semi-Markov sequences and processes, with applications
Authors: Ball Frank
Published: July 2016

In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.
On the time-dependent occupancy and backlog distributions for the GI/G/∞ queue
Authors: Ayhan H. , Limon-Robles J. , Wortman M. A.
Published: July 2016

We consider an infinite server queueing system. An examination of sample path dynamics allows a straightforward development of integral equations having solutions that give time-dependent occupancy (number of customers) and backlog (unfinished work) distributions (conditioned on the time of the first arrival) for the GI/G/∞ queue. These integral equations are amenable to numerical evaluation and can be generalized to characterize GI

X
/G/∞ queue. Two examples are given to illustrate the results.
Inequalities for the score constant in matching random sequences
Authors: Chow Yunshyong , Zhang Yu
Published: July 2016

We consider a sequence matching problem involving the optimal alignment score for contiguous sequences; rewarding matches and penalizing for deletions and mismatches. Arratia and Waterman conjectured in [1] that the score constant a(μ, δ) is a strictly monotone function (i) in δ for all positive δ and (ii) in μ if 0 ≤ μ ≤ 2δ. Here we prove that (i) is true for all δ and (ii) is true for some μ.
On an entropy conservation principle
Authors: Manuceau Jérôme , Troupé Marylène , Vaillant Jean
Published: July 2016

We present an entropy conservation principle applicable to either discrete or continuous variables which provides a useful tool for aggregating observations. The associated method of modality grouping transforms a variable Z
1 into a new variable Z
2 such that the mutual information I(Z
2,Y) between Y, a variable of interest, and Z
2 is equal to I(Z
1,Y).
Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments
Authors: Wang Han-Xing , Fang Dafan
Published: July 2016

A population-size-dependent branching process {Z

n
} is considered where the population's evolution is controlled by a Markovian environment process {ξ
n
}. For this model, let m

k,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Z

n
(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of lim
n

Z

n
and is studied in the case where supθ|m

k,θ − m
θ| → 0 for some real numbers {m
θ} such that infθ
m
θ > 1. When the environmental sequence {ξ
n
} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.
On maximum family size in branching processes
Authors: Rahimov Ibrahim , Yanev George P.
Published: July 2016

The number Y

n
of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Y

n
as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y

n
and EY

n
provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.
An iterative approximation scheme for repetitive Markov processes
Authors: Tüfekçi Tolga , Güllü Refik
Published: July 2016

Repetitive Markov processes form a class of processes where the generator matrix has a particular repeating form. Many queueing models fall in this category such as M/M/1 queues, quasi-birth-and-death processes, and processes with M/G/1 or GI/M/1 generator matrices. In this paper, a new iterative scheme is proposed for computing the stationary probabilities of such processes. An infinite state process is approximated by a finite state process by lumping an infinite number of states into a super-state. What we call the feedback rate, the conditional expected rate of flow from the super-state to the remaining states, given the process is in the super-state, is approximated simultaneously with the steady state probabilities. The method is theoretically developed and numerically tested for quasi-birth-and-death processes. It turns out that the new concept of the feedback rate can be effectively used in computing the stationary probabilities.
A probabilistic analogue of the mean value theorem and its applications to reliability theory
Authors: Di Crescenzo Antonio
Published: July 2016

In a similar spirit to the probabilistic generalization of Taylor's theorem by Massey and Whitt [13], we give a probabilistic analogue of the mean value theorem. The latter is shown to be useful in various contexts of reliability theory. In particular, we provide various applications to the evaluation of the mean total profits of devices having random lifetimes, to the mean total-time-on-test at an arbitrary order statistic of a random sample of lifetimes, and to the mean maintenance cost for the second room of queueing systems in steady state characterized by two serial waiting rooms.
Stochastic properties of a cumulative damage threshold crossing model
Authors: Ebrahimi Nader
Published: July 2016

In this paper we describe a model for survival functions. Under this model a system is subject to shocks governed by a Poisson process. Each shock to the system causes a random damage that grows in time. Damages accumulate additively and the system fails if the total damage exceeds a certain capacity or threshold. Various properties of this model are obtained. Sufficient conditions are derived for the failure rate (FR) order and the stochastic order to hold between the random lifetimes of two systems whose failures can be described by our proposed model.
Characterizations, stochastic equations, and the Gibbs sampler
Authors: Walker Stephen , Damien Paul
Published: July 2016

We obtain characterizations of densities on the real line and provide solutions to stochastic equations using the Gibbs sampler. Particular stochastic equations considered are of the typeX=dB(X+C) andX=dBX+C.
On a single server queue with negative arrivals and request repeated
Authors: Artalejo J. R. , Gomez-Corral A.
Published: July 2016

There is a growing interest in queueing systems with negative arrivals; i.e. where the arrival of a negative customer has the effect of deleting some customer in the queue. Recently, Harrison and Pitel (1996) investigated the queue length distribution of a single server queue of type M/G/1 with negative arrivals. In this paper we extend the analysis to the context of queueing systems with request repeated. We show that the limiting distribution of the system state can still be reduced to a Fredholm integral equation. We solve such an equation numerically by introducing an auxiliary ‘truncated’ system which can easily be evaluated with the help of a regenerative approach.
Optimal active redundancy allocation in k-out-of-n system
Authors: Mi Jie
Published: July 2016

A k-out-of-n system consisting of n components is one that works if and only if at least k of the n components work. Suppose there are n+r (1 ≤ r ≤ n) components available of which r will be used for active redundancy. From the given n+r components, r components are chosen to be used as active redundancies, and another r components receive active redundancies (i.e. these r components are bolstered). Consider a k-out-of-n system consisting of the r bolstered and the other n – r original components. The problem of which r components should be used for active redundancy, and where to allocate them in order to maximize the lifetime of the resulting k-out-of-n system is studied. It is shown that under the usual stochastic ordering ‘≤st’ the first r weakest components should be used for active redundancy and allocated in reverse order to the next r weakest components.
Sharp upper bounds on perfect retrieval in the Hopfield model
Authors: Bovier Anton
Published: July 2016

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.
Partial orders and minimization of records in a sequence of independent random variables
Authors: Gouet Raúl , San Martín Jaime
Published: July 2016

Given independent random variables X
1,…,X

n
, with continuous distributions F
1,…,F

n
, we investigate the order in which these random variables should be arranged so as to minimize the number of upper records. We show that records are stochastically minimized if the sequence F
1,…,F

n
decreases with respect to a partial order, closely related to the monotone likelihood ratio property. Also, the expected number of records is shown to be minimal when the distributions are comparable in terms of a one-sided hazard rate ordering. Applications to parametric models are considered.
On the occurrence of composite events and clusters of points
Authors: Stefanov Valeri T.
Published: July 2016

We derive explicit closed expressions for the moment generating functions of whole collections of quantities associated with the waiting time till the occurrence of composite events in either discrete or continuous-time models. The discrete-time models are independent, or Markov-dependent, binary trials and the events of interest are collections of successes with the property that each two consecutive successes are separated by no more than a fixed number of failures. The continuous-time models are renewal processes and the relevant events are clusters of points. We provide a unifying technology for treating both the discrete and continuous-time cases. This is based on first embedding the problems into similar ones for suitably selected Markov chains or Markov renewal processes, and second, applying tools from the exponential family technology.
A note on acceptance rate criteria for CLTS for Metropolis–Hastings algorithms
Authors: Roberts G. O.
Published: July 2016

This paper considers positive recurrent Markov chains where the probability of remaining in the current state is arbitrarily close to 1. Specifically, conditions are given which ensure the non-existence of central limit theorems for ergodic averages of functionals of the chain. The results are motivated by applications for Metropolis–Hastings algorithms which are constructed in terms of a rejection probability (where a rejection involves remaining at the current state). Two examples for commonly used algorithms are given, for the independence sampler and the Metropolis-adjusted Langevin algorithm. The examples are rather specialized, although, in both cases, the problems which arise are typical of problems commonly occurring for the particular algorithm being used.
A risky asset model with strong dependence through fractal activity time
Authors: Heyde C. C.
Published: July 2016

The geometric Brownian motion (Black–Scholes) model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.
Identifiability for non-stationary spatial structure
Authors: Perrin Olivier , Meiring Wendy
Published: July 2016

For modelling non-stationary spatial random fields Z = {Z(x) : x∊ℝ
n
, n≥2} a recent method has been proposed to deform bijectively the index space so that the spatial dispersion D(x,y) = var[Z(x)-Z(y)], (x,y)∊ℝ
n
xℝ
n
, depends only on the Euclidean distance in the deformed space through an isotropic variogram γ. We prove uniqueness of this model in two different cases: (i) γ is strictly increasing; (ii) γ(u) is differentiable for u > 0.
On the dependence structure and bounds of correlated parallel queues and their applications to synchronized stochastic systems
Authors: Li Haijun , Xu Susan H.
Published: July 2016

This paper studies the dependence structure and bounds of several basic prototypical parallel queueing systems with correlated arrival processes to different queues. The marked feature of our systems is that each queue viewed alone is a standard single-server queuing system extensively studied in the literature, but those queues are statistically dependent due to correlated arrival streams. The major difficulty in analysing those systems is that the presence of correlation makes the explicit computation of a joint performance measure either intractable or computationally intensive. In addition, it is not well understood how and in what sense arrival correlation will improve or deteriorate a system performance measure.

The objective of this paper is to provide a better understanding of the dependence structure of correlated queueing systems and to derive computable bounds for the statistics of a joint performance measure. In this paper, we obtain conditions on arrival processes under which a performance measure in two systems can be compared, in the sense of orthant and supermodular orders, among different queues and over different arrival times. Such strong comparison results enable us to study both spatial dependence (dependence among different queues) and temporal dependence (dependence over different time instances) for a joint performance measure. Further, we derive a variety of upper and lower bounds for the statistics of a stationary joint performance measure. Finally, we apply our results to synchronized queueing systems, using the ideas combined from the theory of orthant and supermodular dependence orders and majorization with respect to weighted trees (Xu and Li (2000)). Our results reveal how a performance measure can be affected, favourably or adversely, by different types of dependencies.