bluuebunny/crossref_metadata_2025 huggingface dataset:
Convergence in distribution of the multi-dimensional Kohonen algorithm
Authors: Sadeghi Ali A.
Published: July 2016

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that the algorithm is not weak Feller, we show that it is a T-chain, regardless of the dimensionalities of both data space and network and the special shape of the neighborhood function. In addition for the practically important case of the multi-dimensional setting, it is shown that the chain is irreducible and aperiodic. We show that these imply the validity of Doeblin's condition, which in turn ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Furthermore, it is shown that the process is positive Harris recurrent, which enables us to use statistical devices to measure the centrality and variability of the invariant probability measure. Our results cover a wide class of neighborhood functions.
Geometric renewal convergence rates from hazard rates
Authors: Berenhaut Kenneth S. , Lund Robert
Published: July 2016

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.
On the optimal stopping values induced by general dependence structures
Authors: Müller Alfred , Rüschendorf Ludger
Published: July 2016

The optimal stopping value of random variables X
1,…,X

n
depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F
1,…,F

n
. They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.
Some peculiarities of exponential random variables
Authors: Litvak Nelly
Published: July 2016

In this paper we utilize a particular transformation of i.i.d. exponential random variables to derive two distributional identities. Throughout the analysis we discover some peculiar properties of exponentials. We also discuss possible generalizations and applications of the results.
The quasistationary distribution of the stochastic logistic model
Authors: Ovaskainen Otso
Published: July 2016

The stochastic logistic model has been studied in various contexts, including epidemiology, population biology, chemistry and sociology. Among the model predictions, the quasistationary distribution and the mean time to extinction are of major interest for most applications, and a number of approximation formulae for these quantities have been derived. In this paper, previous approximation formulae are improved for two mathematically tractable cases: at the limit of the number of individualsN→ ∞ (with relative error of the approximations of the order𝒪(1/N)), and at the limit of the basic reproduction ratioR0→ ∞ (with relative error of the approximations of the order𝒪(1/R0)). The mathematically rigorous formulae are then extended heuristically for other values ofNandR0> 1.
Testing for long memory in the presence of a general trend
Authors: Giraitis Liudas , Kokoszka Piotr , Leipus Remigijus
Published: July 2016

The paper studies the impact of a broadly understood trend, which includes a change point in mean and monotonic trends studied by Bhattacharyaet al.(1983), on the asymptotic behaviour of a class of tests designed to detect long memory in a stationary sequence. Our results pertain to a family of tests which are similar to Lo's (1991) modifiedR/Stest. We show that both long memory and nonstationarity (presence of trend or change points) can lead to rejection of the null hypothesis of short memory, so that further testing is needed to discriminate between long memory and some forms of nonstationarity. We provide quantitative description of trends which do or do not fool theR/S-type long memory tests. We show, in particular, that a shift in mean of a magnitude larger thanN-½, whereNis the sample size, affects the asymptotic size of the tests, whereas smaller shifts do not do so.
Precise large deviations for the prospective-loss process
Authors: Ng Kai W. , Tang Qihe , Yan Jiaan , Yang Hailiang
Published: July 2016

In this paper, we propose a customer-arrival-based insurance risk model, in which customers' potential claims are described as independent and identically distributed heavy-tailed random variables and premiums are the same for each policy. We obtain some precise large deviation results for the prospective-loss process under a mild assumption on the random index (in our case, the customer-arrival process), which is much weaker than that in the literature.
Urn models and differential algebraic equations
Authors: Higueras I. , Moler J. , Plo F. , San Miguel M.
Published: July 2016

The aim of this paper is to study the distribution of colours, {
X

n
}, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process {
X

n
} is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process {
X

n
}.
An exponential functional of random walks
Authors: Szabados Tamás , Székely Balázs
Published: July 2016

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.
Preservation of generalized bathtub-shaped functions
Authors: Savits Thomas H.
Published: July 2016

Block, Savits and Singh previously presented a preservation result for bathtub-shaped functions. In this paper, we strengthen and extend their results to a class of functions which allow for more than one change of monotonicity. Various applications are also considered.
Mixtures of distributions with increasing linear failure rates
Authors: Block Henry W. , Savits Thomas H. , Wondmagegnehu Eshetu T.
Published: July 2016

Populations of specific components are often heterogeneous and consist of a small number of different sub-populations. For example there are often two groups: defective components with shorter lifetimes and standard components with longer lifetimes. Another heterogeneous population results when components produced by two different manufacturing lines are combined. In either case a mixture results. The resulting population can be described using the statistical concept of a mixture. It is a well-known result that a mixture of distributions with decreasing failure rates has a decreasing failure rate. However, little is known about the monotonicity of a mixture when the various subpopulations have failure rates which are not necessarily decreasing. In this paper we study and attempt to determine the shape as well as the overall behavior of the failure rate of a mixture from two subpopulations each of which has increasing linear failure rate.
Convergence theorems for the lengths of consecutive successes of Markov Bernoulli sequences
Authors: Wang Y. H. , Chang Hsiao-Fen , Chen Shun-Yi
Published: July 2016

The limiting distributions of the sums of the lengths of four different kinds of runs of consecutive successes of Markov Bernoulli sequences are derived. It is shown that the limits are convolutions of several distributions involving the Bernoulli, the geometric and the Poisson or compound Poisson distributions.
SIR epidemics on a Bernoulli random graph
Authors: Neal Peter
Published: July 2016

We consider a generalized stochastic epidemic on a Bernoulli random graph. By constructing the epidemic and graph in unison, the epidemic is shown to be a randomized Reed–Frost epidemic. Hence, the exact final-size distribution and extensive asymptotic results can be derived.
Time spent below a random threshold by a Poisson driven sequence of observations
Authors: Kirmani S. N. U. A. , Wesołowski Jacek
Published: July 2016

The mean and the variance of the time S(t) spent by a system below a random threshold until t are obtained when the system level is modelled by the current value of a sequence of independent and identically distributed random variables appearing at the epochs of a nonhomogeneous Poisson process. In the case of the homogeneous Poisson process, the asymptotic distribution of S(t)/t as t → ∞ is derived.
Nonstationarity in ℝn is second-order stationarity in ℝ2n
Authors: Perrin Olivier , Meiring Wendy
Published: July 2016

We prove that a nonstationary random field indexed by ℝnwith moments at least of order 2 can always be viewed as second-order stationary in ℝ2n.
Average optimal policies in a controlled queueing system with dual admission control
Authors: Lewis Mark E.
Published: July 2016

We consider a controlled M/M/1 queueing system where customers may be subject to two potential rejections. The first occurs upon arrival and is dependent on the number of customers in the queue and the service rate of the customer currently in service. The second, which may or may not occur, occurs immediately prior to the customer receiving service. That is, after each service completion the customer in the front of the queue is assessed and the service rate of that customer is revealed. If the second decision-maker recommends rejection, the customer is denied service with a fixed probability. We show the existence of long-run average optimal monotone switching-curve policies. Further, we show that the average reward is increasing in the probability that the second decision-maker's recommendation of rejection is honored. Applications include call centers with delayed classifications and manufacturing systems when the server is responsible for multiple tasks.
Moment convergence in conditional limit theorems
Authors: Janson Svante
Published: July 2016

Consider a sum ∑1
N
Y

i
of random variables conditioned on a given value of the sum ∑1
N
X

i
of some other variables, where X

i
and Y

i
are dependent but the pairs (X

i
,Y

i
) form an i.i.d. sequence. We consider here the case when each X

i
is discrete. We prove, for a triangular array ((X

ni
,Y

ni
)) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.
Compound Poisson approximation for long increasing sequences
Authors: Chryssaphinou Ourania , Vaggelatou Eutichia
Published: July 2016

Consider a sequence X
1,…,X

n
of independent random variables with the same continuous distribution and the event X

i-r+1 < ⋯ < X

i
of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.
An explicit solution to an optimal stopping problem with regime switching
Authors: Guo Xin
Published: July 2016

We investigate an optimal stopping time problem which arises from pricing Russian options (i.e. perpetual look-back options) on a stock whose price fluctuations are modelled by adjoining a hidden Markov process to the classical Black-Scholes geometric Brownian motion model. By extending the technique of smooth fit to allow jump discontinuities, we obtain an explicit closed-form solution. It gives a non-standard application of the well-known smooth fit principle where the optimal strategy involves jumping over the optimal boundary and by an arbitrary overshoot. Based on the optimal stopping analysis, an arbitrage-free price for Russian options under the hidden Markov model is derived.
On the convergence and limits of certain matrix sequences arising in quasi-birth-and-death Markov chains
Authors: He Qi-Ming , Neuts Marcel F.
Published: July 2016

We study the convergence of certain matrix sequences that arise in quasi-birth-and-death (QBD) Markov chains and we identify their limits. In particular, we focus on a sequence of matrices whose elements are absorption probabilities into some boundary states of the QBD. We prove that, under certain technical conditions, that sequence converges. Its limit is either the minimal nonnegative solution G of the standard nonlinear matrix equation, or it is a stochastic solution that can be explicitly expressed in terms of G. Similar results are obtained relative to the standard matrix R that arises in the matrix-geometric solution of the QBD. We present numerical examples that clarify some of the technical issues of interest.
Random bisection and evolutionary walks
Authors: Bach Eric
Published: July 2016

As models for molecular evolution, immune response, and local search algorithms, various authors have used a stochastic process called the evolutionary walk, which goes as follows. Assign a random number to each vertex of the infinite N-ary tree, and start with a particle on the root. A step of the process consists of searching for a child with a higher number and moving the particle there; if no such child exists, the process stops. The average number of steps in this process is asymptotic, as N → ∞, to log N, a result first proved by Macken and Perelson. This paper relates the evolutionary walk to a process called random bisection, familiar from combinatorics and number theory, which can be thought of as a transformed Poisson process. We first give a thorough treatment of the exact walk length, computing its distribution, moments and moment generating function. Next we show that the walk length is asymptotically normally distributed. We also treat it as a mixture of Poisson random variables and compute the asymptotic distribution of the Poisson parameter. Finally, we show that in an evolutionary walk with uniform vertex numbers, the ‘jumps’, ordered by size, have the same asymptotic distribution as the normalized cycle lengths in a random permutation.
Brownian excursions and Parisian barrier options: a note
Authors: Schröder Michael
Published: July 2016

This paper addresses Paris barrier options, as introduced by G. Kentwell and J. Cornwall at Bankers Trust Australia in the mid-1990s, and their valuation, as developed by Chesnay, Jeanblanc-Picqué and Yor using the Laplace-transform approach. The notion of Paris barrier options is extended so that their valuation becomes possible at any point during their lifespan, and the pertinent Laplace transforms of Chesnay, Jeanblanc-Picqué and Yor are modified when necessary.
The intersite distances between pattern occurrences in strings generated by general discrete- and continuous-time models: an algorithmic approach
Authors: Stefanov Valeri T.
Published: July 2016

The formation of patterns from letters of a finite alphabet is considered. The strings of letters are generated by general discrete- and continuous-time models which embrace as particular cases all models considered in the literature. The letters of the alphabet are identified by the states of either discrete- or continuous-time semi-Markov processes. A new and unifying method is introduced for evaluation of the generating functions of both the intersite distance between occurrences of an arbitrary, but fixed, pattern and the waiting time until the first occurrence of that pattern. Our method also covers in a unified way relevant and important joint generating functions. Furthermore, our results lead to an easy and efficient implementation of the relevant evaluations.
Large and small deviations of a string driven by a two-parameter Gaussian noise white in time
Authors: Caithamer Peter
Published: July 2016

Upper as well as lower bounds for both the large deviations and small deviations of several sup-norms associated with the displacements of a one-dimensional string driven by a Gaussian noise which is white in time and has general spatial covariance are developed.
An inspection–repair–replacement model for a deteriorating system with unobservable state
Authors: Yeh Lam
Published: July 2016

In this paper, an inspection–repair–replacement (IRR) model for a deteriorating system with unobservable state is studied. Assume that the system state can only be diagnosed by inspection and an inspection is imperfect. After inspection, if the system is diagnosed as being in a down state, a minimal repair will be undertaken, otherwise we do nothing. Assume further that the system lifetime is a random variable having increasing failure rate. A feasible IRR policy is studied. An algorithm is then suggested for determining an optimal feasible IRR policy for minimizing the long-run average cost per unit time after a finite-step search.
Imperfect repair models with preventive maintenance
Authors: Li Haijun , Shaked Moshe
Published: July 2016

Brown and Proschan (1983) introduced the imperfect repair model, in which an item, upon failure, is replaced with a new one with probability α, and is minimally repaired with probability 1 − α. In this paper we equip the imperfect repair model with preventive maintenance, and we obtain stochastic maintenance comparisons for the numbers of failures under different policies via a point-process approach. We also obtain some results involving stochastic monotonicity properties of these models with respect to the unplanned complete repair probability α.
On the stochastic domination for batch-arrival, batch-service and assemble-transfer queueing networks
Authors: Economou Antonis
Published: July 2016

Stochastic monotonicity properties for various classes of queueing networks have been established in the literature mainly with the use of coupling constructions. Miyazawa and Taylor (1997) introduced a class of batch-arrival, batch-service and assemble-transfer queueing networks which can be thought of as generalized Jackson networks with batch movements. We study conditions for stochastic domination within this class of networks. The proofs are based on a certain characterization of the stochastic order for continuous-time Markov chains, written in terms of their associated intensity matrices.
Burn-in and covariates
Authors: Ebrahimi Nader
Published: July 2016

Burn-in is a widely used engineering procedure useful for eliminating ‘weak’ items and consequently improving the quality of remaining items. The quality of items can be measured via various performance characteristics. In the present paper we develop new performance criteria for the burn-in method. Our criteria not only take into account the reliability of an item, they also incorporate covariates.
Analysis of generalized processor-sharing systems with two classes of customers and exponential services
Authors: Guillemin Fabrice , Pinchon Didier
Published: July 2016

We derive in this paper closed formulae for the joint probability generating function of the number of customers in the two FIFO queues of a generalized processor-sharing (GPS) system with two classes of customers arriving according to Poisson processes and requiring exponential service times. In contrast to previous studies published on the GPS system, we show that it is possible to establish explicit expressions for the generating functions of the number of customers in each queue without calling for the formulation of a Riemann–Hilbert problem. We specifically prove that the problem of determining the unknown functions due to the reflecting conditions on the boundaries of the positive quarter plane can be reduced to a Poisson equation. The explicit formulae are then used to derive some characteristics of the GPS system (in particular the tails of the probability distributions of the numbers of customers in each queue).
Poisson approximation for (k 1, k 2)-events via the Stein-Chen method
Authors: Vellaisamy P.
Published: July 2016

Consider a sequence of independent Bernoulli trials with success probabilityp. LetN(n;k1,k2) denote the number of times thatk1failures are followed byk2successes among the firstnBernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence ofN(n;k1,k2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained forNk3(n;k1,k2), the number of times thatk1failures followed byk2successes occurk3times successively innBernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.
A Brownian motion with two reflecting barriers and Markov-modulated speed
Authors: Kella Offer , Stadje Wolfgang
Published: July 2016

We consider a Brownian motion with time-reversible Markov-modulated speed and two reflecting barriers. A methodology depending on a certain multidimensional martingale together with some linear algebra is applied in order to explicitly compute the stationary distribution of the joint process of the content level and the state of the underlying Markov chain. It is shown that the stationary distribution is such that the two quantities are independent. The long-run average push at the two barriers at each of the states is also computed.
Bias optimal admission control policies for a multiclass nonstationary queueing system
Authors: Lewis Mark E. , Ayhan Hayriye , Foley Robert D.
Published: July 2016

We consider a finite-capacity queueing system where arriving customers offer rewards which are paid upon acceptance into the system. The gatekeeper, whose objective is to ‘maximize’ rewards, decides if the reward offered is sufficient to accept or reject the arriving customer. Suppose the arrival rates, service rates, and system capacity are changing over time in a known manner. We show that all bias optimal (a refinement of long-run average reward optimal) policies are of threshold form. Furthermore, we give sufficient conditions for the bias optimal policy to be monotonic in time. We show, via a counterexample, that if these conditions are violated, the optimal policy may not be monotonic in time or of threshold form.
On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials
Authors: Fu James C. , Chang Y. M.
Published: July 2016

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.
On the severity of M/M/∞ congested episodes
Authors: Preater J.
Published: July 2016

The Laplace transform is obtained for the joint length and height of an M/M/∞ queue excursion above occupancy level C.
Asymptotic independence and a network traffic model
Authors: Maulik Krishanu , Resnick Sidney , Rootzén Holger
Published: July 2016

The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite-dimensional distributions.
Compound random mappings
Authors: Hansen Jennie C. , Jaworski Jerzy
Published: July 2016

In this paper, we introduce a compound random mapping model which can be viewed as a generalization of the basic random mapping model considered by Ross and by Jaworski. We investigate a particular example, the Poisson compound random mapping, and compare results for this model with results known for the well-studied uniform random mapping model. We show that, although the structure of the components of the random digraph associated with a Poisson compound mapping differs from the structure of the components of the random digraph associated with the uniform model, the limiting distribution of the normalized order statistics for the sizes of the components is the same as in the uniform case, i.e. the limiting distribution is the Poisson-Dirichlet (½) distribution on the simplex {{x

i
} : ∑ x

i
≤ 1, x

i
≥ x

i+1 ≥ 0 for every i ≥ 1}.
Characterizing losses during busy periods in finite buffer systems
Authors: Peköz Erol A. , Righter Rhonda , Xia Cathy H.
Published: July 2016

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ = 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and when ρ > 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, when ρ = 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.
Some memoryless bandit policies
Authors: Peköz Erol A.
Published: July 2016

We consider a multiarmed bandit problem, where each arm when pulled generates independent and identically distributed nonnegative rewards according to some unknown distribution. The goal is to maximize the long-run average reward per pull with the restriction that any previously learned information is forgotten whenever a switch between arms is made. We present several policies and a peculiarity surrounding them.
The asymptotic joint normality of the numbers of upper records, lower records and inversions in a random sequence
Authors: Chao Chern-Ching , Chen Yi-Liang , Cheng Wei-Hou
Published: July 2016

We derive the asymptotic joint normality, by a martingale approach, for the numbers of upper records, lower records and inversions in a random sequence.
Large deviation theorems for weighted sums applied to a geographical problem
Authors: Bonin Olivier
Published: July 2016

A large deviation expansion is used to evaluate the impact of errors in a geographical database on the computation of travel times. We work in the framework of discrete random variables and improve a theorem by Book to solve this problem. Simulations are provided to illustrate the methodology.
Breakage and restoration in recursive trees
Authors: Tetzlaff Guillermo Tomás
Published: July 2016

Uniform sequential tree-building aggregation of n particles is analyzed together with the effect of the avalanche that takes place when a subtree rooted at a uniformly chosen vertex is removed. For large n, the expected subtree size is found to be ≃ logn both for the tree of size n and the tree that remains after an avalanche. Repeated breakage-restoration cycles are seen to give independent avalanches which attain size k(1 ≤ k ≤ n-1) with probability (k(k+1))-1 and restored trees that are recursive.
On the convexity of loss probabilities
Authors: Wolff Ronald W. , Wang Chia-Li
Published: July 2016

For the M/G/c loss system, it is well known that Erlang's loss probability is convex in the number of servers. We extend this result firstly to renewal arrivals and exponential service, then to regenerative arrivals and exponential service, and finally to an arbitrary arrival process with i.i.d. service times that are independent of the arrival process.
Some asymptotic results for the induced selection differential
Authors: Nagaraja H. N.
Published: July 2016

We define induced selection differential and discuss asymptotic distribution theory for this quantity. We also obtain the asymptotic joint distribution of the selection differential and the induced selection differential. These are used as measures of improvement in genetic selection programs. We consider the linear regression model set up in detail to obtain various possible limit laws for the induced selection differential.
Semi-Markov decision processes with polynomial reward
Authors: Rosberg Zvi
Published: July 2016

A semi-Markov decision process, with a denumerable multidimensional state space, is considered. At any given state only a finite number of actions can be taken to control the process. The immediate reward earned in one transition period is merely assumed to be bounded by a polynomial and a bound is imposed on a weighted moment of the next state reached in one transition. It is shown that under an ergodicity assumption there is a stationary optimal policy for the long-run average reward criterion. A queueing network scheduling problem, for which previous criteria are inapplicable, is given as an application.
Further properties for unilateral binary processes
Authors: Galbraith R. F. , Walley D.
Published: July 2016

Results and methods discussed in two previous papers are extended to other cases. Comparison is made with recent work by Pickard (1980) and an earlier conjecture is disproved.
Covering the circle with random arcs of random sizes
Authors: Siegel Andrew F. , Holst Lars
Published: July 2016

Consider the random uniform placement of a finite number of arcs on the circle, where the arc lengths are sampled from a distribution on (0, 1). We provide exact formulae for the probability that the circle is completely covered and for the distribution of the number of uncovered gaps, extending Stevens's (1939) formulae for the case of fixed equal arc lengths. A special class of arc length distributions is considered, and exact probabilities of coverage are tabulated for the uniform distribution on (0, 1). Some asymptotic results for the number of gaps are also given.
Levels of null persistency for Markov chains
Authors: Isaacson Dean , Colwell Peter
Published: July 2016

In the study of null-persistent Markov chains the sequence plays an important role. (See for example, Orey (1971), Athreya and Ney (1980).) In this paper we consider the rate of growth of UN
to ∞ as N → ∞. This rate of growth is related to the rate of growth of as N → ∞ and also to the rate at which as n → ∞.
First-passage times for the partial sums of a sequence of geometric distributions
Authors: Woods A. J.
Published: July 2016

It is shown here that questions about the probability distributions of the partial sums of a sequence of geometric distributions, all with different parameters, can be answered by considering the transition probabilities of a homogeneous Markov chain. The result is applied to the embedded random walk of an epidemic process.
Control of grade sizes in a partially stochastic Markov manpower model
Authors: Davies G. S.
Published: July 2016

A manpower model is examined in which there are fixed promotion rates, no demotions and stochastic wastage. We discuss the probability of attainability of structures and structural paths using recruitment control.
Generalized semi-Markov schemes and open queueing networks
Authors: Barbour A. D.
Published: July 2016

Generalized semi-Markov schemes were devised to give a versatile general model embracing queueing networks and similar systems of practical importance, and they have proved particularly successful in uniting many disparate results on insensitivity. However, it turns out that, although closed queueing networks are expressible as GSMS, open networks are not, and that the insensitivity results for such networks are not therefore strictly within their scope. In this paper, it is shown that, as one might hope, open networks can be realized as limits of a suitable sequence of closed networks in such a way that the insensitivity properties of the GSMS are transferred to the open network in the limit, and thus that open networks too can, in a sense, be considered to be GSMS. However, it appears from the technical nature of the arguments involved that, despite this close relationship between GSMS and open networks, it may nonetheless be simpler to treat them separately when constructing the proofs of theorems.
Recurrence formula and the maximum likelihood estimation of the age in a simple branching process
Authors: Adès M. , Dion J.-P. , Labelle G. , Nanthi K.
Published: July 2016

In this paper, we consider a Bienaymé– Galton–Watson process {Xn
; n ≧ 0; Xn
= 1} and develop a recurrence formula for P(Xn = k), k = 1, 2, ···. The problem of obtaining the maximum likelihood estimate of the age of the process when p
0 = 0 is discussed. Furthermore the maximum likelihood estimate of the age of the process when the offspring distribution is negative binomial (p
0 ≠ 0) is obtained, and a comparison with Stigler's estimator (1970) of the age of the process is made.
On the relation between the type–token and species-area problems
Authors: Brainerd Barron
Published: July 2016

The species-area problem in biology and the type-token problem in literary studies are analogues of one another but have nearly disjoint literatures. Here their relationship is treated, a critique of models used in each is made and Markovian models are provided for two well-known empirical models from the species-area literature. Finally, some of the assumptions underlying each of these models are considered in the light of literary data.
On interarrival times in simple stochastic epidemic models
Authors: Yang Grace , Chiang C. L.
Published: July 2016

The probability distributions of the size and the duration of simple stochastic epidemic models are well known. However, in most instances, the solutions are too complicated to be of practical use. In this note, interarrival times of the infectives are utilized to study asymptotic distributions of the duration of the epidemic for a class of simple epidemic models. A brief summary of the results on simple epidemic models in terms of interarrival times is included.
A connective constant for loop-erased self-avoiding random walk
Authors: Lawler Gregory F.
Published: July 2016

A ‘connective constant' is defined for self-avoiding random walk derived by erasing loops from simple random walk. For d ≧ 5, it is shown that this distribution on n-step self-avoiding paths approaches a uniform distribution in a weak sense.
Non-standard insensitivity
Authors: Henderson W.
Published: July 2016

The paper takes a closer look at some insensitivity results which are not contained in the class of insensitive generalised semi-Markov schemes. Based on a study of a special insensitive queue, a conjecture is proposed which covers all known results on insensitive equilibrium distributions. When insensitive events are also included, a study of the G/G/1 queue shows that the conjecture does not always hold.
Weak convergence of the adjusted range of cumulative sums of exchangeable random variables
Authors: Troutman Brent M.
Published: July 2016

Let be the adjusted range of the cumulative sums of a sequence , where . Weak convergence results for random functions constructed from cumulative sums of {Xs
} are used to obtain the asymptotic distribution and moments of when {Xs
} are exchangeable, or symmetrically dependent, random variables.
The accuracy of the diffusion approximation to the expected time to extinction for some discrete stochastic processes
Authors: Grasman J. , Ludwig D.
Published: July 2016

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.
Adaptive control of M/M/1 queues—continuous-time Markov decision process approach
Authors: Yeh Lam , Thomas L. C.
Published: July 2016

By considering continuous-time Markov decision processes where decisions can be made at any time, we show in the case of M/M/1 queues with discounted costs that there exists a monotone optimal policy among all the regular policies.
On the asymptotic distribution of the size of a stochastic epidemic
Authors: Sellke Thomas
Published: July 2016

For a stochastic epidemic of the type considered by Bailey [1] and Kendall [3], Daniels [2] showed that ‘when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels's assumption that the original number of infectives is ‘small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description.
A limiting distribution for Bernoulli trials with second-order Markov dependence
Authors: Brainerd Barron
Published: July 2016

Let ℰ be an event governed by a homogeneous second-order two-state Markov chain. Assume the steady state has been achieved. Consider Nm
the number of times that ℰ occurs in m successive trials. The limiting distribution of P(Nm = k) is obtained under the condition that mP(ℰ)=λ.
A note on Norton's theorem for queuing networks
Authors: Walrand J.
Published: July 2016

We propose a simple proof of Norton's theorem for multiclass quasireversible networks.
A stability property of the Ewens sampling formula
Authors: Sawyer Stanley
Published: July 2016

An error bound for convergence to the Ewens sampling formula is given where the population size or mutation rate may vary from generation to generation, or the population is not yet at equilibrium. An application is given to a model of Hartl and Campbell about selectively-equivalent subtypes within a class of deleterious alleles, and a theorem is proven showing that the size of the deleterious class stays within bounds sufficient to apply the first result. Generalizations are discussed.
Limit diffusions of some stepping-stone models
Authors: Sato Ken-Iti
Published: July 2016

A Markov chain model of a population consisting of a finite or countably infinite number of colonies withNparticles at each colony is considered. There aredtypes of particle and transition from thenth generation to the (n+ 1)th is made up of three stages: reproduction, migration, and sampling. Natural selection works in the reproduction stage. The limiting diffusion operator (asN→∞) for the proportion of types at colonies is found. Convergence to the diffusion is proved under certain conditions.
On the asymptotic normality of the number of replications of a paired comparison
Authors: Menon V. V. , Indira N. K.
Published: July 2016

Consider the number Xm
of comparisons made in a sequence of comparisons between two opponents, which terminates as soon as one opponent wins m comparisons. The convergence of Xm
to the normal variable is completely characterized. The normal approximations to the probability function and to the distribution function of Xm
are obtained for any sufficiently large m, together with estimates of the errors in these approximations. Similar results are obtained for the negative binomial distribution as well. Finally, some simple estimates of the mean, variance and the incomplete beta function with equal arguments are constructed.
Availability measures for coherent systems of separately maintained components
Authors: Baxter Laurence A.
Published: July 2016

A coherent system of components, the failure pattern of each of which being assumed to be modelled by an alternating renewal process, is considered. Formulae are derived for the probability that the system is available at a series of points and for the expected numbers of failures and repairs of the system in a fixed interval. It is shown how the model can be generalised to permit each component to exhibit partial availability in the failed state.
Maintainability of manpower structures — counterexamples, results and conjectures
Authors: Haigh John
Published: July 2016

The question as to what grade structures are maintainable in Markov manpower models, with or without the constraint that the total size be kept constant, is examined. Known results are collected together, examples and further results are given, and an old conjecture is generalised.
The time taken for a population to grow from size m to size km
Authors: Sudbury Aidan
Published: July 2016

Central limit theorems are given for the time taken for a population to increase by a factor k > 1 for a supercritical process, or to decrease by a factor k < 1 for a subcritical process. The size of errors is investigated so that confidence limits can be given for these times to O (m
½) where m is the population size at t = 0.
Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation
Authors: Isobe Etsuo , Sato Shunsuke
Published: July 2016

In this paper we deal with the Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation. The so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process.
Exact and limiting distribution of sustained maxima
Authors: Canfield E. R. , McCormick W. P.
Published: July 2016

Define a random variable

We refer to Yn
as a sustained maximum. Under the assumption that the Xi
's are i.i.d. the exact distribution of Yn
is obtained. Under certain conditions on the underlying distribution F we obtain weak limit results for the Yn
as well. Also a combinatorial extreme-value problem is solved.
Permutation probabilities for gamma random variables
Authors: Henery Robert J.
Published: July 2016

The order statistics of a set of independent gamma variables, in general not identically distributed, may serve as a basis for ordering players in a hypothetical game. An alternative formulation in terms of negative binomial variables leads to an expression for the probability that the random gammas are in a given order. This expression may contain rather many terms and some approximations are discussed — firstly as the gamma parameters αi
tend to equality with all ni
the same, and secondly when the probability of an inversion is small. In another interpretation the probabilities discussed arise in the statement of confidence limits for the ratios of population variances, and here the inversion probability is small enough usually that lower and upper bounds may be given for the probability that the sample variances occur in their expected order. These bounds are calculated from the probability that two variables are in expected order, and for gamma variables this probability is obtained from the F-distribution.
Extreme-point solutions in Markov decision processes
Authors: Assaf David
Published: July 2016

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.
Characterizations of the geometric distribution by distributional properties
Authors: Zijlstra Mynt
Published: July 2016

Some new characterizations of the geometric distribution are studied. A generalization of the characterization by the well-known ‘lack-of-memory' property is given together with some closely related characterizations. Furthermore the modified geometric distribution is characterized by a distributional property of the difference of two successive order statistics. The latter result extends work of Puri and Rubin (1970). Finally the geometric distribution is characterized by a conditional distribution property of the difference of two arbitrary order statistics, which generalizes a result by Arnold (1980). Some of the results given answer open questions put in earlier papers.