bluuebunny/crossref_metadata_2025 huggingface dataset:
The age of a Galton-Watson population with a geometric offspring distribution
Authors: Klebaner F. C. , Sagitov S.
Published: July 2016

Motivated by the question of the age in a branching population we try to recreate the past by looking back from the currently observed population size. We define a new backward Galton-Watson process and study the case of the geometric offspring distribution with parameter p in detail. The backward process is then the Galton-Watson process with immigration, again with a geometric offspring distribution but with parameter 1-p, and it is also the dual to the original Galton-Watson process. We give the asymptotic distribution of the age when the initial population size is large in supercritical and critical cases. To this end, we give new asymptotic results on the Galton-Watson immigration processes stopped at zero.
Path integrals for continuous-time Markov chains
Authors: Pollett P. K. , Stefanov V. T.
Published: July 2016

This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.
Asymptotic failure rate of a Markov deteriorating system with preventive maintenance
Authors: Mercier Sophie , Roussignol Michel
Published: July 2016

We consider a system with a finite state space subject to continuous-time Markovian deterioration while running that leads to failure. Failures are instantaneously detected. This system is submitted to sequential checking and preventive maintenance: up states are divided into ‘good’ and ‘degraded’ ones and the system is sequentially checked through perfect and instantaneous inspections until it is found in a degraded up state and stopped to allow maintenance (or until it fails). Time between inspections is random and is chosen at each inspection according to the current degradation degree of the system. Markov renewal equations fulfilled by the reliability of the maintained system are given and an exponential equivalent is derived for the reliability. We prove the existence of an asymptotic failure rate for the maintained system, which we are able to compute. Sufficient conditions are given for the preventive maintenance policy to improve the reliability and the asymptotic failure rate of the system. A numerical example illustrates our study.
Characterizations of classes of lifetime distributions generalizing the NBUE class
Authors: Klar Bernhard , Müller Alfred
Published: July 2016

We introduce a new class of lifetime distributions exhibiting a notion of positive ageing, called the ℳ-class, which is strongly related to the well-known ℒ-class. It is shown that distributions in the ℳ-class cannot have an undesirable property recently observed in an example of an ℒ-class distribution by Klar (2002). Moreover, it is shown how these and related classes of life distributions can be characterized by expected remaining lifetimes after a family of random times, thus extending the notion of NBUE. We give examples of ℳ-class distributions by using simple sufficient conditions, and we derive reliability bounds for distributions in this class.
A clustering law for some discrete order statistics
Authors: Sethuraman Sunder
Published: July 2016

Let X
1, X
2, …, X

n
be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.
A further extension of the generalized burn-in model
Authors: Cha Ji Hwan
Published: July 2016

In this paper, the generalized burn-in and replacement model considered by Cha (2001) is further extended to the case in which the probability of Type II failure is time dependent. Two burn-in procedures are considered and they are compared in cases when both the procedures are applicable. Under some mild conditions on the failure rate function r(t) and the Type II failure probability function p(t), the problems of determining optimal burn-in time and optimal replacement policy are considered.
Ruin probabilities with dependent rates of interest
Authors: Cai Jun
Published: July 2016

In this paper, we study ruin probabilities in two generalized risk models. The effects of timing of payments and interest on the ruin probabilities in the models are considered. The rates of interest are assumed to have a dependent autoregressive structure. Generalized Lundberg inequalities for the ruin probabilities are derived by a renewal recursive technique. An illustrative application is given to the compound binomial risk process.
Output analysis of a single-buffer multiclass queue: FCFS service
Authors: Kulkarni V. G. , Glazebrook K. D.
Published: July 2016

We consider an infinite capacity buffer where the incoming fluid traffic belongs to K different types modulated by K independent Markovian on-off processes. The kth input process is described by three parameters: (λ
k
, μ
k
, r

k
), where 1/λ
k
is the mean off time, 1/μ
k
is the mean on time, and r

k
is the constant peak rate during the on time. The buffer empties the fluid at rate c according to a first come first served (FCFS) discipline. The output process of type k fluid is neither Markovian, nor on-off. We approximate it by an on-off process by defining the process to be off if no fluid of type k is leaving the buffer, and on otherwise. We compute the mean on time τ
k

on and mean off time τ
k

off. We approximate the peak output rate by a constant r

k

o so as to conserve the fluid. We approximate the output process by the three parameters (λ
k

o, μ
k

o, r

k

o), where λ
k

o = 1/τ
k

off, and μ
k

o = 1/τ
k

on. In this paper we derive methods of computing τ
k

on, τ
k

off and r

k

o for k = 1, 2,…, K. They are based on the results for the computation of expected reward in a fluid queueing system during a first passage time. We illustrate the methodology by a numerical example. In the last section we conduct a similar output analysis for a standard M/G/1 queue with K types of customers arriving according to independent Poisson processes and requiring independent generally distributed service times, and following a FCFS service discipline. For this case we are able to get analytical results.
Stationary solution to the fluid queue fed by an M/M/1 queue
Authors: Barbot N. , Sericola B.
Published: July 2016

We consider an infinite-capacity buffer receiving fluid at a rate depending on the state of an M/M/1 queue. We obtain a new analytic expression for the joint stationary distribution of the buffer level and the state of the M/M/1 queue. This expression is obtained by the use of generating functions which are explicitly inverted. The case of a finite capacity fluid queue is also considered.
On the equivalence of floating- and fixed-strike Asian options
Authors: Henderson Vicky , Wojakowski Rafał
Published: July 2016

There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating- and fixed-strike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.
Branching processes with deteriorating random environments
Authors: Jagers Peter , Lu Zhunwei
Published: July 2016

We introduce Galton-Watson-style branching processes in random environments which are deteriorating rather than stationary or independent. Some primary results on process growth and extinction probability are shown, and two simple examples are given.
Ageing and stochastic comparisons for a covariate failure model
Authors: Pellerey Franco , Semeraro Patrizia
Published: July 2016

Ebrahimi (2001) proposes an interesting model for assessing a system's reliability, expressing the failure time of the system in terms of a deterioration process and covariates. He also provides illustrative examples and gives some properties of the model. In this note, we give conditions for negative ageing of the system's lifetime, and we correct one of his statements. Moreover, for the lifetimes of two systems of the same kind, some stochastic comparisons are presented.
An identification problem in almost and asymptotically almost periodically correlated processes
Authors: Isokawa Y.
Published: July 2016

Consider a unknown realizable time-invariant linear filter driven by a point process. We are interested in the identification of this system, observing only the output random process. If the process is almost periodically correlated but not periodically correlated, we can identify the filter, using the second-order non-stationary spectrum of the process. We do not require the assumption that the filter is minimum phase.
The rumour process
Authors: Dunstan Ross
Published: July 2016

The general stochastic epidemic model is used as a model for the spread of rumours. Recursive expressions are found for the mean of the final size of each generation of hearers. Simple expressions are found for the generation size and the asymptotic form of its final size in the deterministic model. An approximating process is presented.
Exact distributions of kin numbers in a Galton-Watson process
Authors: Joffe A. , Waugh W. A. O'n.
Published: July 2016

The kin number problem involves the relationship between sibship sizes and offspring numbers, and also numbers of relatives of other degrees of affinity of a random member of a population, to be called Ego. The problem has been well known to demographers for some time, but results obtained only gave expected numbers. Recently a study of it, based on the Galton-Watson process, was made, with a view to obtaining joint distributions (Waugh (1981)). In the latter study it was assumed that the population was large, and thus some of the results obtained were approximations.

In the present paper exact distributions are obtained, for any size of population. This can be of use in applications, where the population considered may be a small, isolated tribe or other special group. As a theoretical investigation, it replaces some heuristic arguments with limiting properties that are intrinsic to the process and it makes it possible to evaluate the previous approximations.
The M/G/k group-arrival group-departure loss system
Authors: Fakinos D.
Published: July 2016

This paper considers the equilibrium behaviour of the M/G/k group-arrival group-departure loss system. Such a system has k servers whose customers arrive in groups, the arrival epochs of groups being points of a Poisson process. The duration of a service can be characteristic of the group size; however, customers who belong to the same group have equal service times. The customers of a group start being served immediately upon their arrival, unless their number is greater than the number of idle servers. In this case the whole group leaves and does not return later (i.e. is lost). Among other things, a generalization of the Erlang B-formula is given and it is shown that the arrival and departure processes are statistically indistinguishable.
On the non-existence of consistent estimates in Galton-Watson processes
Authors: Lockhart Richard
Published: July 2016

Estimation of the offspring distribution from a single realization of a supercritical Galton-Watson process is studied. It is shown that, based on population totals, a parameter of the offspring distribution cannot be estimated unless it is determined by the mean, variance, lattice size and lattice offset of the offspring distribution.
Connectivity of random graphs
Authors: Dorea Chang C. Y.
Published: July 2016

We consider a random field {Xij, i, j = 1, ···, n} where the random variables Xij
takes on values 1 or 0. The collection {Xij
} can be viewed as a random graph with nodes {1, ···, n} by interpreting X
ij = 1 as the existence of an arc emanating from the node i to the node j. Such a representation will enable us to study ordered and unordered graphs, being also the general representation of a random graph. In this note the probability that the graph is connected is computed under the condition that Σ
iXki=l for k = 1, · ··, n. This result extends Ross's recent theorems on connectivity of random graphs.
Population-size-dependent branching process with linear rate of growth
Authors: Klebaner F. C.
Published: July 2016

The process we consider is a binary splitting, where the probability of division, , depends on the population size, 2i. We show that Zn
converges to ∞ almost surely on a set of positive probability. Zn
/n converges in distribution to a proper limit, diverges almost surely on converges almost surely on and there are no constants cn
such that Zn
/cn
converges in probability to a non-degenerate limit.
The proportion of quadrilaterals formed by random lines in a plane
Authors: Tanner J. C.
Published: July 2016

An expression is derived for the proportion of quadrilaterals among the polygons formed by random lines in the plane. Its value is 0.381 466….
A note on the geometric/probabilistic relationship in a Markov manpower model
Authors: Davies G. S.
Published: July 2016

The probability of attaining a goal structure in a partially stochastic manpower model is discussed in relation to the position of the structure described in Euclidean space. It is found that, within the regions of structures attainable with non-zero probability, the probabilities decrease on lines emanating from the structure attainable with probability 1.
On the moments of the cumulative idle time in an M/D/∞ queue
Authors: Foley R. D. , Yadin M.
Published: July 2016

Let I denote the fraction of time that an M/D/∞ queue is idle during [0, t]. We give a simple proof that the nth moment of I conditioned on k arrivals equals the kth moment of I conditioned on n arrivals.
The total progeny of a simple branching process with state-dependent immigration
Authors: Kulkarni M. V. , Pakes A. G.
Published: July 2016

The paper establishes limit theorems for the total progeny of a simple branching process in which immigration is allowed whenever the population size reaches 0.
Limit theorems for some sequential occupancy problems
Authors: Janson Svante
Published: July 2016

Consider n cells into which balls are thrown at random until all but m cells contain at least l + 1 balls each. Asymptotic results when n →∞, m and l held fixed, are given for the number of cells containing exactly k balls and for related random variables.
Job search: the continuous case
Authors: Zuckerman Dror
Published: July 2016

In this paper we examine a continuous model of job search. An individual is seeking employment continuously over time. There is a search cost of c monetary units per unit time. Job offers are received randomly over time according to a renewal process. The wage offers are assumed to be positive, independent and identically distributed random variables. The only decision the searcher must make is when to stop searching and accept an offer. The objective is to determine a stopping rule that is optimal under the following optimality criteria:

(a)

maximum expected net return,

(b)

maximum expected discounted net return.

We investigate the case in which the stream of job offers is determined by a Poisson process. We show that under the two optimality criteria defined above an optimal stopping rule is determined by a single critical value ξ. The optimal strategy is to accept the first job offer that exceeds ξ. In the second part of the article, the search model in which offers are received according to a general renewal process is examined.
On a generalized finite-capacity storage model
Authors: Woolford Samuel W.
Published: July 2016

This paper considers a finite-capacity storage model defined on a Markov chain {Xn
; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If Xn
= j, then there is a random ‘input' Vn
(j) (a negative input implying a demand) of ‘type' j, having a distribution function Fj
(·). We assume that {Vn
(j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn
} and of {Vn
(k)}, for k ≠ j. Here, the random variables Vn
(j) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (Zn, Xn
) and (Zn, Qn, Ln
), where Zn
(defined in (1.2)) is the level of storage at time n, Qn
(defined in (1.3)) is the cumulative overflow at time n, and Ln
(defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn
) is obtained.
Limit laws for the maxima of a class of quasi-stationary sequences
Authors: Turkman K. F. , Walker A. M.
Published: July 2016

A class of quasi-stationary sequences of random variables is introduced. After giving the definition, it is shown that the limiting distributions of the maxima of such sequences, when suitably normalized, converge to one of the three extreme-value distributions.
On restricted and pseudo-contagious occupancy distributions
Authors: Charalambides Ch. A.
Published: July 2016

Consider m distinguishable urns with k distinguishable cells each and suppose that n indistinguishable balls are randomly allocated into these urns. When the capacity of each cell is limited to one ball (restricted occupancy) or unlimited (pseudo-contagious occupancy) and empty urns are not permitted, the probability function and factorial moments of the number Mt
= Mt
(n, m, k) of urns containing exactly t balls are expressed in terms of numbers related to the Stirling numbers. The limiting distributions of Mt = Mt
(n, m, k) when k → 0 or k →∞ are derived; these distributions are expressed in terms of the Stirling numbers of the first and second kind respectively. Moreover a modification of the occupancy models of Barton and David is shown to lead to the preceding distributions.
Learning models with continuous time parameter and multivariate point processes
Authors: Pruscha Helmut
Published: July 2016

The concept of a learning model (or random system with complete connections) with continuous time parameter is introduced on the basis of the notion of a multivariate point process possessing an intensity. The stepwise transition probabilities in terms of the intensity are derived and a Monte Carlo method for simulating a sample is presented. By modelling the intensity process various types of learning models can be built. We propose a linear learning model which comprises the continuous-time Markov process as well as Hawkes's mutually exciting point process. We study the asymptotic behaviour of this linear model in terms of explosion or extinction and of convergence of some estimates. We close with some numerical results from computer simulations.
Majorisation, randomness and some discrete distributions
Authors: Hickey Raymond J.
Published: July 2016

Majorisation is used to compare discrete distributions in terms of randomness or, equivalently, informativeness. Some properties of measures of randomness are discussed. The effect on degree of randomness of a change in parameter value is investigated for some distributions including the binomial and Poisson.
Evolutionarily stable strategies of diploid populations with semi-dominant inheritance patterns
Authors: Cressman R. , Hines W. G. S.
Published: July 2016

The assumption of arbitrary and biologically implausible inheritance patterns in sexual diploid populations can yield population models in which convergence of a population's mean strategy to an evolutionarily stable strategy will not occur, even though this strategy is attainable with the correct choice of gametic frequencies. The present paper investigates the effect of imposing a simplifying and biologically reasonable restriction on the assumed inheritance patterns; namely, that dominance or underdominance occurs.
Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process
Authors: Yoshida Minoru
Published: July 2016

Before some random moment θ, independent identically distributed random variables x
1, · ··, xθ–

1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ

+1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's.

In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.
On the determination of a distribution by its median residual life function: a functional equation
Authors: Gupta Ramesh C. , Langford Eric S.
Published: July 2016

In reliability studies, it is well known that the mean residual life function determines the distribution function uniquely. In this paper we show how closely we can determine a distribution when its median residual life function M[S | t] is known. This amounts to solving the functional equation , where R is the reliability function. We actually study a more general functional equation f(φ(t)) = sf(t) called Schroder's equation. It is shown that, under mild assumptions on φ, the solution is of the form f(t) = f
0(t)k(log f
0 (t)), where f
0 is a well-behaved particular solution which can be constructed and k is a periodic function; thus the solution is not unique. Two examples are solved to illustrate the method. Finally, these examples are used to solve the problem of linear M[S | t] studied by Schmittlein and Morrison. As an extra benefit, all of our results hold equally well for the more general sth percentile residual life function.
The asymptotic behavior o queues with time-varying arrival rates
Authors: Heyman Daniel P. , Whitt Ward
Published: July 2016

This paper discusses the asymptotic behavior of the Mt/G/c queue having a Poisson arrival process with a general deterministic intensity. Since traditional equilibrium does not always exist, other notions of asymptotic stability are introduced and investigated. For the periodic case, limit theorems are proved complementing Harrison and Lemoine (1977) and Lemoine (1981).
A note on some results of Schuh
Authors: Doney R. A.
Published: July 2016

We extend recent results of Schuh on the convergence of , where α ≧ 0 and Sn
is the sum of n i.i.d. positive random variables to sums of the form for a large class of functions g, give simpler proofs than those of Schuh, and derive reformulations of the explosion criteria for Markov branching processes with discrete and continuous state space.
On the transition density of a multidimensional parameter Wiener process with one barrier
Authors: Cabaña E. M.
Published: July 2016

An upper estimate of the probability density ofd-parameter Wiener process with one barrier, already known ford= 2, is extended for generald.
On a problem of the busy-period determination in queues with infinitely many servers
Authors: Dvurečenskij A. , Kuljukina L. A. , Ososkov G. A.
Published: July 2016

In this paper the probability distribution of the discrete busy period of the 〈M, 1, GI/∞, 1〉 queue and some of its stability properties are given.
The busy period of order n in the GI/D/∞ queue
Authors: Dvurečenskij A.
Published: July 2016

The busy period of the GI/D/∞ queue is determined as the time when at least one customer is served. Let v be the number of customers served during this period. The busy period of order n is defined as a busy period for which v ≧ n. In this paper we derive the exact distributions, integral equations, characteristic functions and all moments of those periods. Finally, some properties of the idle periods of order n are established.
The central limit theorem for the Poisson shot-noise process
Authors: Lane John A.
Published: July 2016

The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.
On an integral equation for first-passage-time probability densities
Authors: Ricciardi L. M. , Sacerdote L. , Sato S.
Published: July 2016

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.
Conditions for extinction in certain bisexual Galton–Watson branching processes
Authors: Hull David M.
Published: July 2016

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.
Bounds for the probability of complete intersection of random chords in a circle
Authors: Gates John
Published: July 2016

Upper and lower bounds are obtained for the integral giving the probability that n invariant chords to a circle all intersect each other.
Divergent population processes and mammal outbreaks
Authors: Holgate P.
Published: July 2016

Birth and death processes, i.e. Markov chains {X(t)}, 0 ≦t≦ ∞, taking values in the setIof non-negative integers, for which the infinitesimal transition probabilities arehave found wide application in mathematical ecology . In order that the infinite system of differential Equations (1) should have solutionssuch thatPij(t) ≦ 0, alltandit is necessary and sufficient that. The conditions ensure that with probability one, the sample paths ofX(t) have no discontinuities worse than jumps, and in this case the solution is unique. If (3) does not hold then for somet1at least, there is a set of sample paths of positive probability, each of which hast1as a limit point of jumps. For pure birth processes, (in which μj= 0) this clearly impliesX(t)→ ∞ ast→t1.
Epidemic process on a random graph: some preliminary results
Authors: Gertsbakh I. B.
Published: July 2016

A random graph is a collection of n points and n directed arcs: a directed arc goes equiprobably from each point to one of (n – 1) other points. m points are initially ‘infected'. We consider several schemes of epidemic process, e.g. when the infection is delivered according to arc direction. We prove that the probability of infecting all the n points with m = 1 is ∼ e/n, when n → ∞; another result is that m = o(√ n) cannot infect an essential part of the graph (having the size of O(n)). Possible applications of the models to real world phenomena are briefly discussed.
A Markov construction for a multidimensional point process
Authors: Isham Valerie
Published: July 2016

A sequence of finite point processes {Pn
} is constructed in using a Markov sequence of points. Essentially, in the process Pn
consisting of n events, the coordinates of these events are simply the first n points of a Markov sequence suitably scaled so that the average density of the process is independent of n. The second-order properties of Pn
are discussed and sufficient conditions are found for Pn
to converge in distribution to a Poisson process as n →∞. A simple example involving the cardioid distribution is described.
On limit distributions for one- and two-sample Kolmogorov–Smirnov type statistics
Authors: Geller Nancy L.
Published: July 2016

Several new limit theorems for one- and two-sample statistics of the Kolmogorov-Smirnov type are obtained. These generalize results of Takács, Vincze, and others. The proofs combine combinatorial and weak convergence techniques.
Lanchester square-law battles: Transient and terminal distributions
Authors: Goldie Charles M.
Published: July 2016

For a battle governed by the stochastic version of Lanchester's square law the transient distribution of state is found; also various terminal distributions. The terminal distributions arise also in certain related random walks.
A conditional expectation formula for diffusion processes
Authors: Aase Knut K.
Published: July 2016

In some applications of diffusion theory it is of interest to study the expected time of the first exit from an interval given that a specified boundary will be the first exitpoint. This problem is considered here for a regular Feller process, and applications are given.
Chernoff's theorem in the branching random walk
Authors: Biggins J. D.
Published: July 2016

If Fn∗
is the n-fold Stieltjes convolution of the increasing function F, then a version of Chernoff's theorem, on the limiting behaviour of (Fn∗
(na))1/n
, is established for Fn∗
. If Z
(n)(t) is the number of the nth-generation people to the left of t in a supercritical branching random walk then an analogous result is proved for Z
(n).
A note on record times
Authors: Westcott Mark
Published: July 2016

This note indicates two alternative proofs for a result of Galambos and Seneta [1] on the distribution of ratios of successive record times in a sequence of independent and identically distributed random variables. Some consequences of one method of proof are discussed, and its analogues in a continuous time setting are described.
On a class of non-Markov decision processes
Authors: Glazebrook K. D.
Published: July 2016

The optimal strategy for a class of non-Markov decision processes is characterised and has the property that changes of action may occur between successive transitions of the process. Results are given which enable the optimal strategy to be computed iteratively.
The probability that two samples in the plane will have disjoint convex hulls
Authors: Rogers L. C. G.
Published: July 2016

Suppose given an absolutely continuous distribution on the plane, and points P
i, · · ·, Pj
, Πt, · · ·, Πk chosen independently according to the given distribution. Denoting by Gj
the convex hull of {P
i, · · ·, Pj
}, and by Γk the convex hull of (Πt, · · ·, Π
k
}, and writing pjk
for the probability that Gj
and Γ
k
are disjoint, certain properties of the array {pjk
; j, k = 1,2, · · ·} are established, including a recurrence generating the array in terms of {p
1n
; n = 1,2, · · ·}, and asymptotic results for {pnn
; n = 1,2, · · ·}. Some examples are considered.
On calculating extinction probabilities for branching processes in random environments
Authors: Wilkinson William E.
Published: July 2016

Consider a discrete time Markov chain {Zn
} whose state space is the non-negative integers and whose transition probability matrix ║Pij
║ possesses the representation

where {Pr
}, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z
0 = k, a finite positive integer.
Negative binomial processes
Authors: Barndorff-Nielsen Ole , Yeo G. F.
Published: July 2016

This paper is concerned with negative binomial processes which are essentially mixed Poisson processes whose intensity parameter is given by the sum of squares of a finite number of independently and identically distributed Gaussian processes. A study is made of the distribution of the number of points of a k-dimensional negative binomial process in a compact subset of Rk
, and in particular in the case where the underlying Gaussian processes are independent Ornstein-Uhlenbeck processes when more detailed results may be obtained.
Extreme value theory for a class of discrete distributions with applications to some stochastic processes
Authors: Anderson C. W.
Published: July 2016

Let ξn
be the maximum of a set of n independent random variables with common distribution function F whose support consists of all sufficiently large positive integers. Some of the classical asymptotic results of extreme value theory fail to apply to ξn
for such F and this paper attempts to find weaker ones which give some description of the behaviour of ξn
as n → ∞. These are then applied to the extreme value theory of certain regenerative stochastic processes.
Optimal reinsurance
Authors: Dayananda P. W. A.
Published: July 2016

The fundamental principle underlying insurance is that the expected value of claims is equal to the premium. This was established by Bernoulli [4] in 1738. Subsequent work on the development of ‘risk theory’ led to research concerning the probability of ‘ruin’ of an insurance company. A critical study of these investigations was made by Borch [3] in a recent paper.
Random displacements of regularly spaced events
Authors: Nelsen Roger B. , Williams Trevor
Published: July 2016

We consider the point process i + τ
i
(i = 0, ± 1, ± 2, . . .), where the τ
i
(assumed for convenience to be positive) are independent random samples from the same distribution. We define an inter-arrival interval as the stretch between neighbouring events, which need not belong to successive values of i, because of the jumbling of position occasioned by the variability of the τ. We obtain explicit expressions for the distribution of inter-arrival intervals in general, as well as the correlation coefficient between successive inter-arrival intervals in the case where the τ are exponentially distributed.
Bonferroni bounds revisited
Authors: Kounias Stratis , Sotirakoglou Kiki
Published: July 2016

Lower and upper bounds of degree m for the probability of the union of n not necessarily exchangeable events are established. These bounds may be constructed to improve the Bonferroni and the Sobel–Uppuluri bounds.

An application to equi-correlated multivariate normal distribution is given.
Insensitivity of processes with interruptions
Authors: Henderson W. , Taylor P.
Published: July 2016

The theory of insensitivity within generalised semi-Markov processes is extended to cover classes of models in which the generally distributed lifetimes can be terminated prematurely by the deaths of negative exponentially distributed lifetimes. As a consequence of this approach it is shown that there exist classes of processes which are insensitive with respect to characteristics of the general distributions other than the mean. Two examples are given. The first is an analysis of networks of queues in which the generally distributed service times can be interrupted with resulting changes in routing probabilities. The second is a model for the effect of disturbances on the evolution of a vegetation community.
On the dependence structure of hitting times of multivariate processes
Authors: Ebrahimi Nader , Ramallingam T.
Published: July 2016

A direct approach to derive dependence properties among the hitting times of bivariate processes has been initiated by Ebrahimi (1987) and explored further by Ebrahimi and Ramallingam (1988). In this paper, new results are obtained for multivariate processes, which help us to identify positive and negative dependence structures among the hitting times of the processes. Applications of our theorems to reliability of systems are given.
Infinite-dimensional stochastic difference equations for particle systems and network flows
Authors: Darling R. W. R.
Published: July 2016

Let V be a countably infinite set, and let {Xn, n = 0, 1, ·· ·} be random vectors in which satisfy Xn
= AnXn

– 1 + ζ
n
, for i.i.d. random matrices {An
} and i.i.d. random vectors {ζ
n
}. Interpretation: site x in V is occupied by Xn
(x) particles at time n; An
describes random transport of existing particles, and ζ n
(x) is the number of ‘births' at x. We give conditions for (1) convergence of the sequence {Xn
} to equilibrium, and (2) a central limit theorem for n–1/2(X
1 + · ·· + Xn
), respectively. When the matrices {An
} consist of 0's and 1's, these conditions are checked in two classes of examples: the ‘drip, stick and flow model' (a stochastic flow with births), and a neural network model.
Multichannel queueing systems with infinite waiting room and stochastic control
Authors: Dshalalow Jewgeni
Published: July 2016

A wide class of multichannel queueing models appears to be useful in practice where the input stream of customers can be controlled at the moments preceding the customers' departures from the source (e.g. airports, transportation systems, inventories, tandem queues). In addition, the servicing facility can govern the intensity of the servicing process that further improves flexibility of the system. In such a multichannel queue with infinite waiting room the queueing process {Zt
; t ≧ 0} is under investigation. The author obtains explicit formulas for the limiting distribution of (Zt
) partly using an approach developed in previous work and based on the theory of semi-regenerative processes. Among other results the limiting distributions of the actual and virtual waiting time are derived. The input stream (which is not recurrent) is investigated, and distribution of the residual time from t to the next arrival is obtained. The author also treats a Markov chain embedded in (Zt
) and gives a necessary and sufficient condition for its existence. Under this condition the invariant probability measure is derived.
Stochastic monotonicity in general queueing networks
Authors: Shanthikumar J. George , Yao David D.
Published: July 2016

We develop a representation for general queueing networks, and study some stochastic monotonicity properties that have been previously established in Jackson networks (e.g. Shanthikumar and Yao (1986), (1987)).
Extremal processes, secretary problems and the 1/e law
Authors: Pfeifer Dietmar
Published: July 2016

We consider a class of secretary problems in which the order of arrival of candidates is no longer uniformly distributed. By a suitable embedding in a time-transformed extremal process it is shown that the asymptotic winning probability is again 1/e as in the classical situation. Extensions of the problem to more than one choice are also considered.
Stability of extremes with random sample size
Authors: Voorn W. J.
Published: July 2016

A non-degenerate distribution function F is called maximum stable with random sample size if there exist positive integer random variables Nn, n = 1, 2, ···, with P(Nn
= 1) less than 1 and tending to 1 as n → ∞ and such that F and the distribution function of the maximum value of Nn
independent observations from F (and independent of Nn
) are of the same type for every index n. By proving the converse of an earlier result of the author, it is shown that the set of all maximum stable distribution functions with random sample size consists of all distribution functions F satisfying

where c
2, c
3, · ·· are arbitrary non-negative constants with 0 < c2
+ c3
+ · ·· <∞, and all distribution functions G and H defined by F(x)= G(c + exp(x)) and F(x) = H(c – exp(–x)), –∞ < x <∞, where c is an arbitrary real constant.
n-point, win-by-k games
Authors: Siegrist Kyle
Published: July 2016

Consider a sequence of Bernoulli trials between players A and B in which player A wins each trial with probability p∈ [0, 1]. For positive integers n and k with k ≦ n, an (n, k) contest is one in which the first player to win at least n trials and to be ahead of his opponent by at least k trials wins the contest. The (n, 1) contest is the Banach match problem and the (n, n) contest is the gambler's ruin problem. Many real contests (such as the World Series in baseball and the tennis game) have an (n, 1) or an (n, 2) format. The (n, k) contest is formulated in terms of the first-exit time of the graph of a random walk from a certain region of the state-time space. Explicit results are obtained for the probability that player A wins an (n, k) contest and the expected number of trials in an (n, k) contest. Comparisons of (n, k) contests are made in terms of the probability that the stronger player wins and the expected number of trials.
Invariance principles in queueing theory
Authors: Alex Michael , Steinebach Josef
Published: July 2016

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk
}and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.
On the busy period distribution of the M/G/2 queueing system
Authors: Wiens Douglas P.
Published: July 2016

Equations are derived for the distribution of the busy period of the GI/G/2 queue. The equations are analyzed for the M/G/2 queue, assuming that the service times have a density which is an arbitrary linear combination, with respect to both the number of stages and the rate parameter, of Erlang densities. The coefficients may be negative. Special cases and examples are studied.
Some new results on the subexponential class
Authors: Murphree Emily S.
Published: July 2016

A distribution function F on (0,∞) belongs to the subexponential class if the ratio of 1 – F
(2)(x) to 1 – F(x) converges to 2 as x →∞. A necessary condition for membership in is used to prove that a certain class of functions previously thought to be contained in has members outside of . Sufficient conditions on the tail of F are found which ensure F belongs to ; these conditions generalize previously published conditions.
Random polymorphisms and random evolutionarily stable strategies: a comparison
Authors: Haigh John
Published: July 2016

Recently, the properties of ESSs for random payoff matrices, and stable polymorphisms for random fitness matrices, have been investigated. These problems are very closely related, and some similarities and differences are examined here. In contrast to some earlier work,non-robustness of the conclusions, to changes in the distribution of the random matrices, are found in two areas: in the asymptotic number of polymorphisms (or ESSs) whose support is a given size, and in the location of a polymorphism (or ESS) whose size of support is 2 or 3, in a large matrix.
Optimal stopping problems with generalized objective functions
Authors: Hill T. P. , Kennedy D. P.
Published: July 2016

Optimal stopping of a sequence of random variables is studied, with emphasis on generalized objectives which may be non-monotone functions ofEXt, wheretis a stopping time, or may even depend on the entire vector (E[X1I{t=l}], · ··,E[XnI{t=n}]),such as the minimax objective to maximize minj{E[XjI{t=j}]}.Convexity is used to establish a prophet inequality and universal bounds for the optimal return, and a method for constructing optimal stopping times for such objectives is given.
On starr and vardi's estimates of the number of transmission sources
Authors: Hall Peter
Published: July 2016

We propose an estimator for the number of sources, N, which transmit signals according to a Poisson process with a common rate μ. This estimation problem was originated by Starr and pursued by Vardi. Our procedure is a ‘double sampling’ one and has the advantage that it can be used when μ is unknown.
Filtering counts of cars entering and leaving a parking place
Authors: Andersen Søren , Rudemo Mats
Published: July 2016

Cars entering and leaving a parking place in a forest are registered automatically in connection with an investigation of forest recreational use. By subtraction the number of cars parked is computed. Due to erroneous and missing counts, errors occur and may for instance indicate a negative number. The theory of incompletely observed Markov chains is used to obtain an estimate of the number of cars parked.
On a sufficient condition for superprocesses due to whittle
Authors: Glazebrook K. D.
Published: July 2016

Whittle has given a sufficient condition for a certain class of Markov decision processes to have an optimal strategy of a particularly simple form. We show this condition to be necessary in a certain sense and give some examples of its application.
On the mean number of random digits until a given sequence occurs
Authors: Blom Gunnar
Published: July 2016

Random digits are collected one at a time until a given sequence is obtained. The mean number of digits until this happens is determined. Generalizations are given to a stopping region consisting of several sequences of equal length.
On a theorem of bingham and doney
Authors: De Meyer A.
Published: July 2016

We obtain an extension of a theorem of Bingham and Doney connecting the random variables Z
1 and W in the supercritical Galton-Watson process. The regular variation of the distribution of Z1 is equivalent to the regular variation of the tail of the distribution of W for integer values of α > 1.
The stable pedigrees of critical branching populations
Authors: Nerman Olle
Published: July 2016

The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.
A multitrap model for high-intensity reciprocity failure in the photographic grain
Authors: Anderson William J.
Published: July 2016

The response of the photographic grain to light is a non-deterministic process which is as yet not completely understood. This response, as measured by the photographic density, is usually taken to be a function of the product of incident light intensity and exposure time interval duration, but at extreme values of either of these two quantities, this is no longer true. This latter effect is called reciprocity-law failure.

This paper discusses a probabilistic model, similar to a multiserver queue, for high-intensity reciprocity failure.
Radial generation of n-dimensional poisson processes
Authors: Quine M. P. , Watson D. F.
Published: July 2016

A simple method is proposed for the generation of successive ‘nearest neighbours' to a given origin in ann-dimensional Poisson process. It is shown that the method provides efficient simulation of random Voronoi polytopes. Results are given of simulation studies in two and three dimensions.
On the random coverage of the circle
Authors: Holst L. , Hüsler J.
Published: July 2016

Place n arcs of equal length a uniformly at random on the circumference of a circle. We discuss the joint limit distributions of the number of gaps, the uncovered proportion of the circle and the lengths of the largest gap and of the smallest gap, depending on how a → 0 as n →∞.

We show that the results may be proved in a unified and simple way by using a result of Le Cam.
Some models for multitype spatial point processes, with remarks on analysing multitype patterns
Authors: Lotwick H. W.
Published: July 2016

Two classes of ergodic stationary multitype spatial point processes are constructed. These processes have the property that interactions between the types exist, but cannot be detected using standard second-order methods of analysis. Simulations indicate that the interactions can, however, be detected by using ‘empty space' techniques.