bluuebunny/crossref_metadata_2025 huggingface dataset:
A comparison of the ordinary and a varying parameter exponential smoothing
Authors: Bonsdorff Heikki
Published: July 2016

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn
= ß(Zn –

1)Xn
+(1 – ß (Zn

–
1))Zn

–
1, where Xn
are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 < c < d < 1.

Together with (Zn
), we consider the ordinary exponential smoothing Yn
= αXn
+ (1 – α)Yn –

1 where α is a constant, 0 < α < 1. We show that (Yn
) and (Zn
) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn
converge to limits EY, EZ,
respectively, with a geometric convergence rate. Moreover, we show that Ez
is strictly less than EY
= EXn.
Circuit-switching networks with priorities on a one-dimensional lattice
Authors: Shamsiev Rahim
Published: July 2016

We prove existence and uniqueness of a stationary regime for a model of a circuit-switching network with infinitely many nodes.
The rendezvous problem on discrete locations
Authors: Anderson E. J. , Weber R. R.
Published: July 2016

Two friends have become separated in a building or shopping mall and and wish to meet as quickly as possible. There are n possible locations where they might meet. However, the locations are identical and there has been no prior agreement where to meet or how to search. Hence they must use identical strategies and must treat all locations in a symmetrical fashion. Suppose their search proceeds in discrete time. Since they wish to avoid the possibility of never meeting, they will wish to use some randomizing strategy. If each person searches one of the n locations at random at each step, then rendezvous will require n steps on average. It is possible to do better than this: although the optimal strategy is difficult to characterize for general n, there is a strategy with an expected time until rendezvous of less than 0.829 n for large enough n. For n = 2 and 3 the optimal strategy can be established and on average 2 and 8/3 steps are required respectively. There are many tantalizing variations on this problem, which we discuss with some conjectures.
On point processes on the circle
Authors: Hüsler Jürg
Published: July 2016

Point processes on the circle with circumference 1 are considered, which are related to the coverage problem of the circle by n randomly placed arcs of a fixed length. The anticlockwise endpoint of each arc is assumed to be uniformly distributed on the circle. We deal with a general limit result on the convergence of these point processes to a Poisson process on the circle. This result is then applied to several cases of the coverage problem, giving improved limit results in these cases. The proof uses a new convergence result of general point processes.
Optimal stopping in a continuous search model
Authors: Zuckerman Dror
Published: July 2016

We examine a continuous search model in which rewards (e.g. job offers in a search model in the labor market, price offers for a given asset, etc.) are received randomly according to a renewal process determined by a known distribution function. The rewards are non-negative independent and have a common distribution with finite mean. Over the search period there is a constant cost per unit time. The searcher's objective is to choose a stopping time at which he receives the highest available reward (offer), so as to maximize the net expected discounted return. If the interarrival time distribution in the renewal process is new better than used (NBU), it is shown that the optimal stopping strategy possesses the control limit property. The term ‘control limit policy' refers to a strategy in which we accept the first reward (offer) which exceeds a critical control level ξ.
Benes's formula for M/G/1–FIFO ‘explained' by preemptive-resume LIFO
Authors: Cooper Robert B. , Niu Shun-Chen
Published: July 2016

We provide a term-by-term interpretation of Beneš's well-known but mysterious inversion of the Pollaczek–Khintchine formula. The strategy is to recognize the equality of waiting time in M/G/1–FIFO with remaining work in M/G/1–LIFO–preemptive resume. In the process, we give a new and simple derivation of some known results for M/G/1–LIFO–preemptive resume.
An improved Poisson limit theorem for sums of dissociated random variables
Authors: Barbour A. D. , Eagleson G. K.
Published: July 2016

A Poisson limit theorem for sums of dissociated 0–1 random variables is refined by deriving the first terms in an asymptotic expansion. The most natural refinement does not remove all the first-order error in a number of applications to tests of clustering, and a further approximation is derived which gives excellent results in practice. The proofs are based on the technique of Stein and Chen.
Stochastic inequalities for an overflow model
Authors: Hordijk Arie , Ridder Ad
Published: July 2016

A general method to obtain insensitive upper and lower bounds for the stationary distribution of queueing networks is sketched. It is applied to an overflow model. The bounds are shown to be valid for service distributions with decreasing failure rate. A characterization of phase-type distributions with decreasing failure rate is given. An approximation method is proposed. The methods are illustrated with numerical results.
Reliability importance for continuum structure functions
Authors: Kim Chul , Baxter Laurence A.
Published: July 2016

A continuum structure function is a non-decreasing mapping from the unit hypercube to the unit interval. A definition of the reliability importance, ℛi(α) say, of component i at system level α(0 < α ≦ 1) is proposed. Some properties of this function are deduced, in particular conditions under which and conditions under which ℛi(α) is positive (0 < α < 1).
Two theorems on solutions of differential-difference equations and applications to epidemic theory
Authors: Severo Norman C.
Published: July 2016

We present two theorems that provide simple iterative solutions of special systems of differential-difference equations. We show as examples of the theorems the simple stochastic epidemic (cf. Bailey, 1957, p. 39, and Bailey, 1963) and the general stochastic epidemic (cf. Bailey, 1957; Gani, 1965; and Siskind, 1965), in each of which we let the initial distribution of the number of uninfected susceptibles and the number of infectives be arbitrary but assume the total population size bounded. In all of the references cited above the methods of solution involve solving a corresponding partial differential equation, whereas we deal directly with the original system of ordinary differential-difference equations. Furthermore in the cited references the authors begin at time t = 0 with a population having a fixed number of uninfected susceptibles and a fixed number of infectives. For the simple stochastic epidemic with arbitrary initial distribution we provide solutions not obtainable by the results given by Bailey (1957 or 1963). For the general stochastic epidemic, if we use the results of Gani or Siskind, then the solution of the problem having an arbitrary initial distribution would involve additional steps that would sum proportionally-weighted conditional results.
On nonlinear processes involving population growth and diffusion
Authors: Montroll Elliott W.
Published: July 2016

A number of years ago, R. A. Fisher discussed the problem of the propagation of a virile mutant in a population. At about the same time, Kolmogorov, Petrovsky, and Piscounoff, whom we shall refer to as KPP, investigated a general class of partial differential equations which describe simultaneous growth and diffusion processes.
Multivariate normal integrals for highly correlated samples from a wiener process
Authors: McFadden J. A. , Lewis James L.
Published: July 2016

If X
1, …, Xn
obey a multivariate normal distribution with zero means, then the probability that Xi > a for all i = 1, …, n is often called a multivariate normal integral. Such integrals have been considered by various investigators, particularly when a = 0. (For a bibliography, see Gupta.)
Interaction between migration and breeding studied by means of genetic algebras
Authors: Holgate P.
Published: July 2016

In this paper the theory of genetic algebras introduced by Etherington [3], [5] is developed to cover the case where the population is divided into a number of partially isolated subpopulations, between which a limited amount of migration is permitted. It is known [3], [6], [7], [8], [10], that in the absence of selective factors most genetic segregation systems can be represented by algebras belonging to the class of special train algebras, discussed in detail in [4].
Transient behavior of multi-server queues with recurrent input and exponential service times
Authors: Bhat U. Narayan
Published: July 2016

Customers arrive in a recurrent process and get served by one of the s (≧1) servers wit han exponential service time distribution. The equilibrium behavior of the queue length process has been studied by earlier authors. In this paper the transient behavior of this process is investigated.
On probability properties of measures of random sets and the asymptotic behavior of empirical distribution functions
Authors: Ailam Gedalia
Published: July 2016

Probability properties of the measure of the union of random sets have theoretical as well as practical importance (David (1950), Garwood (1947), Hemmer (1959)). In the present paper we derive asymptotic properties of the distributions of these measures and apply the derived properties to the investigation of the asymptotic behavior of empirical distribution functions. Thus, an asymptotic distribution function for the relative lengths of steps in the empirical distribution function is obtained.
A theory of dams with continuous input and a general release rule
Authors: Moran P.A.P.
Published: July 2016

Consider an infinite capacity dam in which the input and release occur continuously in time. Write X(t) for the total input up to time t starting from X(0) = 0 at t = 0. Let Z(t) be the content of the dam at time t and R(u) (0 ≦ u < ∞) a release function such that in any interval of time (t, t + dt), the amount of water released is R(Z(t))dt + o(t) for any bounded realisation of the process {Z(t)}. Thus R(u) can be regarded as a “rate of release”.
Sums of Lifetimes in Age Dependent Branching Processes
Authors: Weiner J. Howard
Published: July 2016

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σ
k=0
∞
pk

s
k and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.
Estimation for a family of life distributions with applications to fatigue
Authors: Birnbaum Z.W. , Saunders S.C.
Published: July 2016

The estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More important, the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.
Recurrence times of clusters of Poisson points
Authors: Leslie R.T.
Published: July 2016

In a previous paper (Leslie (1967)) the distribution of recurrence times for a particular set of success-failure patterns on a sequence of Bernoulli trials was investigated. We now consider the analogous events in continuous time and obtain the Laplace Transform (L.T.) of the distribution of recurrence times; numerical inversion yields the distribution functions.
Spitzer's condition for asymptotically symmetric random walk
Authors: Doney R. A.
Published: July 2016

If the step-length distribution function F for a random walk {Sn, n ≧ 0} is either continuous and symmetric or belongs to the domain of attraction of a symmetric stable law, then it is clear that the symmetric form of ‘Spitzer's condition' holds, i.e.

The question considered in this note is whether or not (⋆) can hold for other random walks. The answer is in the affirmative, for we show that (⋆) holds for a large class of random walks for which F is neither symmetric nor belongs to any domain of attraction; all such random walks are asymptotically symmetric, in the sense that lim
x→∞ {F(–x)| 1 – F(x)} = 1, but we show by an example that this is not a sufficient condition for (⋆) to hold.
Maximizing the intersection density of fibre processes
Authors: Janson Svante , Kallenberg Olav
Published: July 2016

It is shown that the specific intersection density of a stationary Poisson process of cylindrical fibres in R
d, d ≧ 2, attains its maximum when the directions are uniformly distributed. This generalizes a result by R. Davidson for line processes in R
2.
A note on the cost of the simple epidemic
Authors: Jerwood D.
Published: July 2016

The concept of cost has not, until recently, been investigated in any detail within the framework of epidemic theory. In the case of minor illnesses a pertinent estimate of cost may be the man-hours lost in industry during an outbreak. For the general epidemic, in which the removals are immune recoveries (or deaths), this is conveniently represented by the area generated by the trajectory in time t of infectives I(t) until the epidemic ceases. A small element of area under the trajectory is I(t)δt and represents I(t) infectives withdrawn from the labour force for a small time δt; the total work-time lost is thus represented by where T is the duration of the epidemic.
Limit theorems for sequences of jump Markov processes approximating ordinary differential processes
Authors: Kurtz T. G.
Published: July 2016

In [3] this author gave conditions under which a sequence of jump Markov processes Xn
(t) will converge to the solution X(t) of a system of first order ordinary differential equations, in the sense that

for every δ > 0.
A characterization of triangular arrays which are expectations of order statistics
Authors: Kadane Joseph B.
Published: July 2016

Necessary and sufficient conditions are found for a triangular array (finite or infinite) to be expected values of order statistics. The conditions are a wellknown recurrence relation, and a moment condition.
Linear growth in near-critical population-size-dependent multitype Galton–Watson processes
Authors: Klebaner Fima C.
Published: July 2016

We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. We identify cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.
Packing densities of randomly constructed codes
Authors: Sutton Clifton
Published: July 2016

Codes having all pairs of words separated by a Hamming distance of at least d are stochastically constructed by sequentially packing randomly generated q-ary n-tuples. Estimates of the random packing densities are obtained by repeated simulation. Using non-linear regression to fit the estimated densities, an asymptotic approximation formula is obtained for the packing densities which depends only on q, n, d, and an empirical constant.
The hazard and vitality measures of ageing
Authors: Kupka Joseph , Loo Sonny
Published: July 2016

A new measure of the ageing process called the vitality measure is introduced. It measures the ‘vitality' of a time period in terms of the increase in average lifespan which results from surviving that time period. Apart from intrinsic interest, the vitality measure clarifies the relationship between the familiar properties of increasing hazard and decreasing mean residual life. The main theorem asserts that increasing hazard is equivalent to the requirement that mean residual life decreases faster than vitality. It is also shown for general (i.e. not necessarily absolutely continuous) distributions that the properties of increasing hazard, increasing failure rate, and increasing probability of ‘sudden death' are all equivalent.
A note on the multitype Markov branching process
Authors: Gadag V. G.
Published: July 2016

We consider a supercritical, p-dimensional Markov branching process (MBP). Based on the finite and the infinite lines of descent of particles of this p-dimensional MBP, we construct an associated 2p-dimensional process. We show that such a process is a 2p-dimensional, supercritical MBP. This 2p-dimensional process retains the branching property when conditioned on the sets of extinction and non-extinction. Asymptotic results and central limit theorems for the associated process and the original process are established by using the results of Gadag and Rajarshi (1987).
The number of extreme points in the convex hull of a random sample
Authors: Aldous David J. , Fristedt Bert , Griffin Philip S. , Pruitt William E.
Published: July 2016

Let {Xk} be an i.i.d. sequence taking values in ℝ2with the radial and spherical components independent and the radial component having a distribution with slowly varying tail. The number of extreme points in the convex hull of {X1, · ··,Xn} is shown to have a limiting distribution which is obtained explicitly. Precise information about the mean and variance of the limit distribution is obtained.
On the critical infection rate of the one-dimensional basic contact process: numerical results
Authors: Ziezold Herbert , Grillenberger Christian
Published: July 2016

Instead of the basic contact process on with infection rate λ we consider for m ≧ 0 the Markov process starting with ξ
0(k) = 1 for k ≧ 0 and ξ
0(k)= 0 for k < 0 and with changing only those k which are at most m places to the right of the left-most infected cell. For m = 0, 1,· ··, 14 direct computations give critical values which are lower bounds for the critical value of the original basic contact process.
First-passage-time density and moments of the ornstein-uhlenbeck process
Authors: Ricciardi Luigi M. , Sato Shunsuke
Published: July 2016

A detailed study of the asymptotic behavior of the first-passage-time p.d.f. and its moments is carried out for an unrestricted conditional Ornstein-Uhlenbeck process and for a constant boundary. Explicit expressions are determined which include those earlier discussed by Sato [15] and by Nobile et al. [9]. In particular, it is shown that the first-passage-time p.d.f. can be expressed as the sum of exponential functions with negative exponents and that the latter reduces to a single exponential density as time increases, irrespective of the chosen boundary. The explicit expressions obtained for the first-passage-time moments of any order appear to be particularly suitable for computation purposes.
Hitting times of decreasing sets for regenerative processes
Authors: Athreya K. B.
Published: July 2016

Let {X(t): t ≧ 0} be a regenerative process with excursion process {Y(u): 0 ≦ u < T, T} and state space . Let An
be a sequence of sets in such that pn
=P(Y(u)∊An
for some 0≦ u < T) → 0. Let Vn
be the hitting time of An
for the process X. This paper gives a variety of conditions on the excursion process Y to obtain limit theorems for Vn. Apart from obtaining an exponential limit in the positive recurrent case, i.e. ET < x, some non-exponential limits are obtained in the null recurrent case. The results are illustrated via the age process.
Modelling the flow of coalescing data streams through a processor
Authors: Anantharam V.
Published: July 2016

In a data processing network, two data streams A and B arrive at a node independently at the same Poisson rate λ. Service at exponential rate µ can take place iff there is at least one of each of A and B present. The output is the combined processed data AB. We consider models of this situation with finite buffers, with infinite buffers and with finite buffers for the excess of each input type over the other. We apply the filtering theory for point process functionals of a Markov chain to study whether the output flow is Poisson in equilibrium. The motivation is to examine, if the output is to subsequently be processed by a queueing system, whether it can be treated as an independent Poisson input to that system.

A result of independent interest is that a subset of transitions of a countable-state Markov process does not yield a Poisson process when counted, if the rate matrix of counted transitions is nilpotent, and we prove a generalization of Pakes' lemma for countable-state Markov chains.
Stereological analysis of particles of varying ellipsoidal shape
Authors: Møller J.
Published: July 2016

Stereological analysis of d-dimensional particles of ellipsoidal shape based on lower-dimensional sections through the particles is discussed. It is proved that the non-void intersections between three parallel hyperplanes and an ellipsoid uniquely determine the ellipsoid, and based on this fact we may reconstruct ellipsoids from sectional information. Combining this reconstruction with a new sampling procedure we obtain a useful tool for non-parametric stereological analysis of particle aggregates of ellipsoids. Finally, parametric models for ellipsoids which are mathematically convenient for the present set up are introduced and discussed.
On Cox processes and gamma renewal processes
Authors: Yannaros Nikos
Published: July 2016

It is shown that the gamma distribution with shape parameter α can be obtained through a p-thinning for every 0 < p < 1, when 0 < α ≦ 1. In the case α > 1, the gamma distribution cannot be obtained through thinning. The class of renewal processes with gamma-distributed times between events is considered. It is shown that an ordinary gamma renewal process is a Cox process if and only if 0 < α ≦ 1. Necessary and sufficient conditions for delayed gamma renewal processes to be Cox are also given. Finally, a short description of the gamma renewal process as a Cox process is given.
Construction of first-passage-time densities for a diffusion process which is not necessarily time-homogeneous
Authors: Jáimez R. Gutiérrez , Gonzalez A. Juan , Román P. Román
Published: July 2016

In Giorno et al. (1988) a new method for constructing first-passage-time probability density functions is outlined. This rests on the possibility of constructing the transition p.d.f. of a new time-homogeneous diffusion process in terms of a preassigned transition p.d.f. without making use of the classical space-time transformations of the Kolmogorov equation (Ricciardi (1976)).

In the present paper we give an extension of this result to the case of a diffusion process X(t) which is not necessarily time-homogeneous, and a few examples are presented.
Sequential limits in Markov set-chains
Authors: Hartfiel D. J.
Published: July 2016

Let T be a non-empty subset of η X n stochastic matrices. Define

The sequence T
1, T
2, · ·· is called a Markov set-chain. An important problem in this area is to determine when such a set-chain converges. This paper gives a notion of a sequential limiting set and shows how it can be used to obtain a result on set-chain convergence.
An optimal stopping problem in the excavation of archaeological remains
Authors: Nakai Teruhisa
Published: July 2016

In the excavation of archaeological remains of high value, a search tool with low cost can be used at first, but if we reach a stratum which is very likely to contain valuable material we must use a search tool with high efficiency to reduce the risk of destruction of the remains even if this is more expensive. Which tool should we use at each time? Furthermore, if the search costs too much in comparison with the reward of detection, it is not profitable to go on searching for ever. When should we stop the search? We answer these questions for continuous and discrete models.
The Mk
/M/∞ queue with heterogeneous customers in a batch
Authors: Choi Bong Dae , Park Kwang Kyu
Published: July 2016

We consider the Mk
/M/∞ queue with k heterogeneous customers in a batch where the customer of type i in a batch requires an exponential service time with parameter µi. In steady state, the joint generating function of the number of customers of type i being served in the system is derived explicitly by solving a partial differential equation.
Search models
Authors: Bather J. A.
Published: July 2016

Mathematical models have been proposed for oil exploration and other kinds of search. They can be used to estimate the amount of undiscovered resources or to investigate optimal stopping times for the search. Here we consider a continuous search for hidden objects using a model which represents the number and values of the objects by mixtures of Poisson processes. The flexibility of the model and its complexity depend on the number of components in the mixture. In simple cases, optimal stopping rules can be found explicitly and more general qualitative results can sometimes be obtained.
An insensitivity property of ladder height distributions
Authors: Frenz Michael , Schmidt Volker
Published: July 2016

This paper considers the undershoot of a general continuous-time risk process with dependent increments under a certain initial level. The increments are given by the locations and amounts of claims which are described by a stationary marked point process. Under a certain balance condition, it is shown that the distribution of the undershoot depends only on the mark distribution and on the intensity of the underlying point process, but not on the form of its distribution. In this way an insensitivity property is extended which has been proved in Björk and Grandell [3] for the ruin probability, i.e. for the probability that after a finite time interval the initial level will be crossed from above.
New models for Markov random fields
Authors: Cressie Noel , Lele Subhash
Published: July 2016

The Hammersley–Clifford theorem gives the form that the joint probability density (or mass) function of a Markov random field must take. Its exponent must be a sum of functions of variables, where each function in the summand involves only those variables whose sites form a clique. From a statistical modeling point of view, it is important to establish the converse result, namely, to give the conditional probability specifications that yield a Markov random field. Besag (1974) addressed this question by developing a one-parameter exponential family of conditional probability models. In this article, we develop new models for Markov random fields by establishing sufficient conditions for the conditional probability specifications to yield a Markov random field.
Multivariate imperfect repair
Authors: Sheu Shey-Huei , Griffith William S.
Published: July 2016

We consider models of systems of components with dependent life-lengths having certain multivariate distributions. Upon failure, components are repaired. Two types of repair are possible. After perfect repair, a unit has the same life distribution as a new item. After imperfect repair, a unit has the life distribution of an item which is of the same age but has never failed. Different sources of failure are distinguished and affect the probabilities of perfect and imperfect repair.
Optimality of index policies for stochastic scheduling with switching penalties
Authors: Van Oyen Mark P. , Pandelis Dimitrios G. , Teneketzis Demosthenis
Published: July 2016

We investigate the impact of switching penalties on the nature of optimal scheduling policies for systems of parallel queues without arrivals. We study two types of switching penalties incurred when switching between queues: lump sum costs and time delays. Under the assumption that the service periods of jobs in a given queue possess the same distribution, we derive an index rule that defines an optimal policy. For switching penalties that depend on the particular nodes involved in a switch, we show that although an index rule is not optimal in general, there is an exhaustive service policy that is optimal.
Optimal policies for machine repairmen problems
Authors: Frostig Esther
Published: July 2016

n unreliable machines are maintained by m repairmen. Assuming exponentially distributed up-time and repair time we find the optimal policy to allocate the repairmen to the failed machines in order to stochastically minimize the time until all machines work. Considering only one repairman, we find the optimal policy to maximize the expected total discount time that machines work. We find the optimal policy for the cases where the up-time and repair time are exponentially distributed or identically arbitrarily distributed up-times and increasing failure rate distribution repair times.
A random allocation model for carrier-borne epidemics
Authors: Daley D. J. , Gani J.
Published: July 2016

A carrier-borne epidemic is considered in which the carriers, subject to a death process, infect susceptibles by random allocation rather than the classical homogeneous mixing process. An explicit solution for the probability generating function (p.g.f.) of the process is obtained, and a probabilistic analysis of carrier models provided. The sizes and durations of the random allocation and classical carrier epidemics are compared. The strongest comparisons concern sample path results based on the probabilistic analysis; this also gives a sounder basis for computational work.
Optimal foraging of a reproducing animal as a discounted reward problem
Authors: Merad S. , McNamara J. M.
Published: July 2016

A model in which a foraging animal can reproduce if its energy reserves reach a critical level is presented. The animal's reserves are modelled as a finite-state Markov chain. The animal has the choice between foraging options which may have the same mean net gain but differ in their variances. In addition to starvation, animals are subject to death because of predation, bad weather, and so on. We focus on the case where the rate of mortality due to these sources is the same under all options. We investigate policies that maximise the expected lifetime reproductive success. It is found that the optimal value function is concave at low reserves (low-variance action region) and convex at high reserves (high-variance action region). The value function under the low-variance action has also the same shape and the same inflexion point. This result allows us to compute optimal policies just by looking at the low-variance value function. The result is also used to show that increasing the mortality rate increases the high-variance region under the optimal policy. The pattern of risk-sensitive behaviour predicted by this model is in contrast to that predicted by a similar model in which no reproduction occurs and the optimality criterion is to minimise the probability of death (McNamara (1990)).
A Dam with seasonal input
Authors: Lund Robert B.
Published: July 2016

This paper examines the infinitely high dam with seasonal (periodic) Lévy input under the unit release rule. We show that a periodic limiting distribution of dam content exists whenever the mean input over a seasonal cycle is less than 1. The Laplace transform of dam content at a finite time and the Laplace transform of the periodic limiting distribution are derived in terms of the probability of an empty dam. Necessary and sufficient conditions for the periodic limiting distribution to have finite moments are given. Convergence rates to the periodic limiting distribution are obtained from the moment results. Our methods of analysis lean heavily on the coupling method and a stochastic monotonicity result.
Asymptotic behaviour of Markov population processes by asymptotically linear rate of change
Authors: Klebaner F. C.
Published: July 2016

Multidimensional Markov processes in continuous time with asymptotically linear mean change per unit of time are studied as randomly perturbed linear differential equations. Conditions for exponential and polynomial growth rates with stable type distribution are given. From these conditions results on branching models of populations with stabilizing reproduction for near-supercritical and near-critical cases follow.
Exact aggregation of absorbing Markov processes using the quasi-stationary distribution
Authors: Ledoux James , Rubino Gerardo , Sericola Bruno
Published: July 2016

We characterize the conditions under which an absorbing Markovian finite process (in discrete or continuous time) can be transformed into a new aggregated process conserving the Markovian property, whose states are elements of a given partition of the original state space. To obtain this characterization, a key tool is the quasi-stationary distribution associated with absorbing processes. It allows the absorbing case to be related to the irreducible one. We are able to calculate the set of all initial distributions of the starting process leading to an aggregated homogeneous Markov process by means of a finite algorithm. Finally, it is shown that the continuous-time case can always be reduced to the discrete one using the uniformization technique.
Generalized adaptive exponential smoothing procedures
Authors: Herkenrath Ulrich
Published: July 2016

The classical exponential smoothing procedure to be applied to a sequence of observations is modified into some adaptive variants and also generalized to the recursive scheme

This means that given the history of observations and smoothed values up to time n, the updated smoothed value Wn +

1 is given by a function u of Wn
and Xn
, which satisfies only certain structural properties. We study these procedures within the framework of random systems with complete connections and exploit results from Markov process theory to study the sequence . A broad class of ‘smoothing functions' u, which induce sequences with nice statistical properties, is presented. A method to estimate the limit of the expected smoothed values is also developed.,
Uniqueness of maximum values in discrete distributions
Authors: Floréen Patrik , Huuskonen Taneli
Published: July 2016

We discuss the probability that among a number of random discrete values there is a unique maximal value. Some general inequalities are derived. As an application, we study the limit behavior of the probability of uniqueness for a subexponential number of binomially distributed variables.
On the local behaviour of ladder height distributions
Authors: Bertoin J. , Doney R. A.
Published: July 2016

There is a well-known connection between the asymptotic behaviour of the tail of the distribution of the increasing ladder height and the integrated tail of the step distribution of a random walk which either drifts to –∞, or oscillates and whose decreasing ladder height has finite mean. We establish a similar connection in a local sense; this means that in the lattice case we link the asymptotic behaviours of the mass function of the ladder height distribution and of the tail of the step distribution. We deduce the asymptotic behaviour of the mass function of the maximum of the walk, when this is finite, and also treat the non-lattice case.
On multiple covering of a circle with random arcs
Authors: Holst Lars
Published: July 2016

On a circle of unit circumference arcs of length a are placed at random. Let N
α
be equal to the necessary number of arcs to cover at least the length 1 − p, 0 ≦ p < 1, of the circumference at least m (≧1) times. In the present paper limit distributions of Nα
are derived when α → 0. Some results for spacings are also obtained.
On the reversible counters system of Lampard
Authors: Phatarfod R. M.
Published: July 2016

The paper considers some properties of the reversible counters system of Lampard [1]. Lampard considers the properties of the process imbedded at the epochs when the counter receiving pulses from the ‘down’ process reaches zero. We consider here the time-dependent and stationary distributions of the original process. Their derivation gives rise to some interesting functional equations, besides being useful for studying queues with Markov-dependent inter-arrival or service times.
The robustness of positive recurrence and recurrence of Markov chains under perturbations of the transition probabilities
Authors: Tweedie Richard L.
Published: July 2016

In many Markov chain models, the immediate characteristic of importance is the positive recurrence of the chain. In this note we investigate whether positivity, and also recurrence, are robust properties of Markov chains when the transition laws are perturbed. The chains we consider are on a fairly general state space : when specialised to a countable space, our results are essentially that, if the transition matrices of two irreducible chains coincide on all but a finite number of columns, then positivity of one implies positivity of both; whilst if they coincide on all but a finite number of rows and columns, recurrence of one implies recurrence of both. Examples are given to show that these results (and their general analogues) cannot in general be strengthened.
Estimation theory for growth and immigration rates in a multiplicative process
Authors: Heyde C. C. , Seneta E.
Published: July 2016

This paper deals with the simple Galton-Watson process with immigration, {Xn} with offspring probability generating function (p.g.f.)F(s) and immigration p.g.f.B(s), under the basic assumption that the process is subcritical (0 <m≡F'(1–) < 1), and that 0 <λ≡B'(1–) < ∞, 0 <B(0) < 1, together with various other moment assumptions as needed. Estimation theory for the ratesmandλon the basis of a single terminated realization of the process {Xn} is developed, in that (strongly) consistent estimators for bothmandλare obtained, together with associated central limit theorems in relation tomandμ≡λ(1–m)–1Following this, historical antecedents are analysed, and some examples of application of the estimation theory are discussed, with particular reference to the continuous-time branching process with immigration. The paper also contains a strong law for martingales; and discusses relation of the above theory to that of a first order autoregressive process.
Convergence of critical Galton-Watson branching processes
Authors: Lindvall Torgny
Published: July 2016

The purpose of this note is to extend results on critical Galton-Watson branching processes, due to Feller and Lamperti: if the number of individuals at time zero tends to infinity, the process converges, with two different normalizations, to certain diffusion processes in the functional form, discussed in, e.g., Billingsley.
A characterization of the negative exponential distribution with application to reliability theory
Authors: Phillips M. J.
Published: July 2016

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.
A piecewise linear stochastic differential equation driven by a Lévy process
Authors: Reed Josh , Zwart Bert
Published: July 2016

We consider a stochastic differential equation (SDE) with piecewise linear drift driven by a spectrally one-sided Lévy process. We show that this SDE has some connections with queueing and storage models, and we use this observation to obtain the invariant distribution.
Dixie cups: sampling with replacement from a finite population
Authors: Klaassen Chris A. J.
Published: July 2016

At which (random) sample size will every population element have been drawn at least m times? This special coupon collector's problem is often referred to as the Dixie cup problem. Some asymptotic properties of the Dixie cup problem with unequal sampling probabilities are described.
Characterizing the idle-period distribution of GI/G/1 queues
Authors: Li Jingwen , Ou Jihong
Published: July 2016

A variety of performance measures of aGI/G/1 queue are explicitly related to the idle-period distribution of the queue, suggesting that the system analysis can be accomplished by the analysis of the idle period. However, the ‘stand-alone' relationship for the idle-period distribution of theGI/G/1 queue (i.e. the counterpart of Lindley's equation) has not been found in the literature. In this paper we develop a non-linear integral equation for the idle period distribution of theGI/G/1 queue. We also show that this non-linear system defines a unique solution. This development makes possible the analysis of theGI/G/1 queue in a different perspective.
The final outcome and temporal solution of a carrier-borne epidemic model
Authors: Ball Frank , Clancy Damian
Published: July 2016

We consider a stochastic model for the spread of a carrier-borne epidemic amongst a closed homogeneously mixing population, in which a proportion 1 − π of infected susceptibles are directly removed and play no part in spreading the infection. The remaining proportion π become carriers, with an infectious period that follows an arbitrary but specified distribution. We give a construction of the epidemic process which directly exploits its probabilistic structure and use it to derive the exact joint distribution of the final size and severity of the carrier-borne epidemic, distinguishing between removed carriers and directly removed individuals. We express these results in terms of Gontcharoff polynomials. When the infectious period follows an exponential distribution, our model reduces to that of Downton (1968), for which we use our construction to derive an explicit expression for the time-dependent state probabilities.
Geometric Markov chains
Authors: Rising William
Published: July 2016

A generalization of the familiar birth–death chain, called the geometric chain, is introduced and explored. By the introduction of two families of parameters in addition to the infinitesimal birth and death rates, the geometric chain allows transitions beyond the nearest neighbor, but is shown to retain the simple computational formulas of the birth–death chain for the stationary distribution and the expected first-passage times between states. It is also demonstrated that even when not reversible, a reversed geometric chain is again a geometric chain.
Poisson and compound Poisson approximations for random sums of random variables
Authors: Vellaisamy P. , Chaudhuri B.
Published: July 2016

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.
Lundberg inequalities for a Cox model with a piecewise constant intensity
Authors: Schmidli Hanspeter
Published: July 2016

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li
) of successive levels of the intensity forms a Markov chain. The duration σi
of the level Li
is assumed to be only dependent via Li
. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li
contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).
Autoregressive processes with infinite variance
Authors: Hannan E. J. , Kanter Marek
Published: July 2016

The least squares estimators β
i(N), j = 1, …, p, from N data points, of the autoregressive constants for a stationary autoregressive model are considered when the disturbances have a distribution attracted to a stable law of index α < 2. It is shown that N1/δ(β
i(N) – β) converges almost surely to zero for any δ > α. Some comments are made on alternative definitions of the βi
(N).
On aggregated Markov processes
Authors: Fredkin Donald R. , Rice John A.
Published: July 2016

A finite-state Markov process is aggregated into several groups. What can be learned about the underlying process from the aggregated one? We provide some partial answers to this question.
A general risk process and its properties
Authors: Scheike Thomas H.
Published: July 2016

We construct a risk process, where the law of the next jump time or jump size can depend on the past through earlier jump times and jump sizes. Some distributional properties of this process are established. The compensator is found and some martingale properties are discussed.
Some problems on a one-dimensional correlated random walk with various types of barrier
Authors: Zhang Yuan Lin
Published: July 2016

In this paper one-dimensional correlated random walks (CRW) with various types of barrier such as elastic barriers, absorbing barriers and so on are defined, and explicit expressions are derived for the ultimate absorbing probability and expected duration. Some numerical examples to illustrate the effects of correlation are also presented.
Les familles exponentielles statistiques invariantes par les groupes du cône et du paraboloide de révolution
Authors: Letac Gérard
Published: July 2016

Statistical exponential families invariant with respect to the groups of the cone or the paraboloid of revolution are discussed. B(x, y) denotes the symmetric bilinear form on x
0
y
0 – x
1
y
1 – ·· ·– xdyd
on ℝ
d+1, C denotes the cone of revolution in ℝ
d+1 {x; B(x,x) > 0 and x
0 > 0}, and, for p > ½(d−1), μ p
is the positive measure on ℝ
d+1 defined by its Laplace transform ⨍ exp (B(x,y))μ
p
(dy) = (B(y,y))−p
for y on C. More precisely, if p > ½ (d−1) one has

and μ½(d−1) concentrated on the boundary ∂C. This paper studies the natural exponential families
On the moments of ladder epochs for driftless random walks
Authors: Chow Yuan S.
Published: July 2016

Let X, X1, X2, … be i.i.d. Sn
=Σ1

n

Xj
, E|X| > 0, E(X) = 0 and τ = inf {n ≥ 1: Sn
≥ 0}. By Wald's equation, E(τ) =∞. If E(X
2) <∞, then by a theorem of Burkholder and Gundy (1970), E(τ
1/2) =∞. In this paper, we prove that if E((X–
)
2
) <∞, then E(τ
1/2) =∞. When X is integer-valued and X ≥ −1 a.s., a necessary and sufficient condition for E(τ
1–1/p
) <∞, p > 1, is Σn–1–1p E|Sn| <∞.
Poisson approximations for the distribution and moments of ordered m-spacings
Authors: Glaz Joseph , Naus Joseph , Roos Malgorzata , Wallenstein Sylvan
Published: July 2016

This article investigates the accuracy of approximations for the distribution of ordered m-spacings for i.i.d. uniform observations in the interval (0, 1). Several Poisson approximations and a compound Poisson approximation are studied. The result of a simulation study is included to assess the accuracy of these approximations. A numerical procedure for evaluating the moments of the ordered m-spacings is developed and evaluated for the most accurate approximation.