bluuebunny/crossref_metadata_2025 huggingface dataset:
A stochastic ordering of partial sums of independent random variables and of some random processes
Authors: Boland Philip J. , Proschan Frank , Tong Y. L.
Published: July 2016

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.
Markov chains and generalized continued fractions
Authors: Hanschke Thomas
Published: July 2016

This paper deals with a class of discrete-time Markov chains for which the invariant measures can be expressed in terms of generalized continued fractions. The representation covers a wide class of stochastic models and is well suited for numerical applications. The results obtained can easily be extended to continuous-time Markov chains.
Convergence theorems for a class of simulated annealing algorithms on ℝd
Authors: Bélisle Claude J. P.
Published: July 2016

We study a class of simulated annealing algorithms for global minimization of a continuous function defined on a subset of We consider the case where the selection Markov kernel is absolutely continuous and has a density which is uniformly bounded away from 0. This class includes certain simulated annealing algorithms recently introduced by various authors. We show that, under mild conditions, the sequence of states generated by these algorithms converges in probability to the global minimum of the function. Unlike most previous studies where the cooling schedule is deterministic, our cooling schedule is allowed to be adaptive. We also address the issue of almost sure convergence versus convergence in probability.
Expected size distributions in models of group dynamics
Authors: Morgan Byron J. T.
Published: July 2016

This paper considers several models for biological processes in which animate individuals live and die as members of groups which can split to form smaller groups. Resulting distributions of individuals over groups are compared and contrasted. In particular, two qualitatively different types of distributions are identified. It is clear that distinguishing between models giving rise to the same distribution types is difficult. Implications for more complex models are discussed and avenues for further research are outlined.
Repair replacement policies
Authors: Block Henry W. , Langberg Naftali A. , Savits Thomas H.
Published: July 2016

In this paper we introduce the concept of repair replacement. Repair replacement is a maintenance policy in which items are preventively maintained when a certain time has elapsed since their last repair. This differs from age replacement where a certain amount of time has elapsed since the last replacement. If the last repair was a complete repair, repair replacement is essentially the same as age replacement. It is in the case of minimal repair that these two policies differ. We make comparison between various types of policies in order to determine when and under which condition one type of policy is better than another.
On the empty cells of Poisson histograms
Authors: Kendall Wilfrid S.
Published: July 2016

This paper considers the histogram of unit cell size built up from m independent observations on a Poisson (μ) distribution. The following question is addressed: what is the limiting probability of the event that there are no unoccupied cells lying to the left of occupied cells of the histogram? It is shown that the probability of there being no such isolated empty cells (or isolated finite groups of empty cells) tends to unity as the number m of observations tends to infinity, but that the corresponding almost sure convergence fails. Moreover this probability does not tend to unity when the Poisson distribution is replaced by the negative binomial distribution arising when μ is randomized by a gamma distribution. The relevance to empirical Bayes statistical methods is discussed.
Orientation-dependent chord length distributions characterize convex polygons
Authors: Nagel W.
Published: July 2016

It is shown that the convex polygons are uniquely determined (up to translation and reflection) by their covariograms. The covariogram can be represented by the ‘orientation-dependent chord length distribution', i.e. the distribution of the length of chords which are generated by random lines parallel to fixed directions. Thus the result contributes to answer Blaschke's question about the content of information comprised in chord length distributions.
Random walks on discrete and continuous circles
Authors: Rosenthal Jeffrey S.
Published: July 2016

We consider a large class of random walks on the discrete circle Z/(n), defined in terms of a piecewise Lipschitz function, and motivated by the ‘generation gap' process of Diaconis. For such walks, we show that the time until convergence to stationarity is bounded independently of n. Our techniques involve Fourier analysis and a comparison of the random walks on Z/(n) with a random walk on the continuous circle S
1.
Absorption and first-passage times for a compound Poisson process in a general upper boundary
Authors: Gallot S. F. L.
Published: July 2016

This paper considers the absorption of a non-decreasing compound Poisson process of finite order in a general upper boundary. The problem is relevant in fields such as risk theory, Kolmogorov–Smirnov statistics and sequential analysis. The probability of absorption and first-passage times are given in terms of a generating function which depends on the boundary only and can be computed readily. Absorption is certain or not as the asymptotic slope of the boundary is greater or less than the expected increase of the process in unit time. The case of the linear boundary is considered in detail.
Notes on random chords in convex bodies
Authors: Enns E. G. , Ehlers P. F.
Published: July 2016

Consider a compact body E embedded in a convex compact body G. A point P is randomly chosen in E and three different rays to the boundary ∂G of G are generated by P. Ray is in a uniformly random direction and has length R, ray is through a second random point chosen from within G and has length W, and ray is to a random point in ∂G and has length Y. The distribution of Y is obtained and is related to previous work. When E is specialized to G or to ∂G, other known results are retrieved. The paper ends with a discussion of a conjecture relating the means of R, W and Y.
A new approach to (s, S) inventory systems
Authors: Hu Jian-Qiang , Nananukul Soracha , Gong Wei-Bo
Published: July 2016

In this paper, we consider period review (s, S) inventory systems with independent and identically distributed continuous demands and full backlogging. Using an approach recently proposed by Gong and Hu (1992), we derive an infinite system of linear equations for all moments of inventory level. Based on this infinite system, we develop two algorithms to calculate the moments of the inventory level. In the first one, we solve a finite system of linear equations whose solution converges to the moments as its dimension goes to infinity. In the second one, we in fact obtain the power series of the moments with respect to s and S. Both algorithms are based on some very simple recursive procedures. To show their efficiency and speed, we provide some numerical examples for the first algorithm.

(s, S) INVENTORY SYSTEMS; DYNAMIC RECURSIVE EQUATIONS; INFINITE LINEAR EQUATIONS; MACLAURIN SERIES
Hitting probabilities for random ellipses and ellipsoids
Authors: Duma Andrei , Stoka Marius
Published: July 2016

Let denote a rectangular lattice in the Euclidean plane E
2, generated by (a × b) rectangles. In this paper we consider the probability that a random ellipse having main axes of length 2α and 2ß, with intersects . We regard the lattice as the union of two orthogonal sets and of equidistant lines and evaluate the probability that the random ellipse intersects or . Moreover, we consider the dependence structure of the events that the ellipse intersects or . We study further the case when the main axes of the ellipse are parallel to the lines of the lattice and satisfy 2ß = min (a, b) < 2α = max (a, b). In this case, the probability of intersection is 1, and there exist almost surely two perpendicular segments in within the ellipse. We evaluate the distribution function, density, mean and variance of the length of these segments. We conclude with a generalization of this problem in three dimensions.
A reversibility relationship for two Markovian time series models by stationary exponential tailed distribution
Authors: Littlejohn R. P.
Published: July 2016

The continuous autoregressive and minification stationary non-negative time series models discussed by Chernick et al. (1988) are generalized to model marginal distributions which have atoms of mass at zero. The reversibility theorem relating these processes with exponential marginal distributions is extended to the case where the marginal distribution has exponential tail.
Lundberg bounds on the tails of compound distributions
Authors: Willmot Gordon E. , Lin Xiaodong
Published: July 2016

Exponential bounds are derived for the tail probabilities of various compound distributions, generalizing the classical Lundberg inequality of insurance risk theory. Failure rate properties of the compounding distribution including log-convexity and log-concavity are considered in some detail. Mixed Poisson compounding distributions are also considered. A ruin theoretic generalization of the Lundberg inequality is obtained in the case where the number of claims process is a mixed Poisson process. An application to the M/G/1 queue length distribution is given.
Functional approximation theorems for controlled renewal processes
Authors: Konstantopoulos Takis , Papadakis Spyros N. , Walrand Jean
Published: July 2016

We prove a functional law of large numbers and a functional central limit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith interarrival time is scaled by a function of the number of previous i arrivals. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solution of an ordinary differential equation. Likewise, the functional central limit theorem establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation. Our proofs are based on martingale and time-change arguments.
A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes
Authors: Goldstein Martin I.
Published: July 2016

Let Z(t) ··· (Z
1(t), …, Zk
(t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v.

It is shown that, under second moment assumptions,

uniformly for s ∈ ([0, 1]
k
of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant.

This result is then used to give a new proof of the exponential limit law.
Uniform bounding of probability generating functions and the evolution of reproduction rates in birds
Authors: Heyde C. C. , Schuh H.-J.
Published: July 2016

Many species of birds have a characteristic clutch size which is either fixed atkor is of the formkork+ 1 for some appropriate integerk.In this paper we show, using a multitype Galton–Watson process to model a bird population, that such behaviour can correspond to maximization of the probability of survival of the species to timetfor each finitet.This is also a conclusion which might be drawn from the theory of natural selection and hence provides some mathematical evidence of the force of evolution. The results of the paper rest on a bounding of probability generating functions.
Sur une caractérisation des vidanges fortes: corrections and extensions
Authors: Dion Jean-Guy
Published: July 2016

A draining is a stochastic process defined by an urn scheme where the successive drawings are made without replacement and according to a drawing algorithm associated with a weighted graph. A draining is said to be a strong one if the number of balls which can be drawn from the urn (under the algorithm) converges in probability to the total number of balls in the urn at the beginning of the drawing when it goes to infinity. In particular, drainings associated with complete graphs having equal weights are found to be strong and for some others associated weighted graphs, strong drainings exist.
Limiting diffusion for random walks with drift conditioned to stay positive
Authors: Kao Peichuen
Published: July 2016

Let {ξ

k
: k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ
1} = μ ≠ 0. Form the random walk {S

n
: n ≧ 0} by setting S
0, S

n
= ξ
1 + ξ
2 + ··· + ξ

n
, n ≧ 1. Define the random function Xn
by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ
1) that the finite-dimensional distributions of Xn
, conditioned on n < N < ∞ converge to those of the Brownian excursion process.
On the maximum and absorption time of left-continuous random walk
Authors: Pakes Anthony G.
Published: July 2016

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.

Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.
Oscillating Brownian motion
Authors: Keilson Julian , Wellner Jon A.
Published: July 2016

An ‘oscillating' version of Brownian motion is defined and studied. ‘Ordinary' Brownian motion and ‘reflecting' Brownian motion are shown to arise as special cases. Transition densities, first-passage time distributions, and occupation time distributions for the process are obtained explicitly. Convergence of a simple oscillating random walk to an oscillating Brownian motion process is established by using results of Stone (1963).
The optimality equation in average cost denumerable state semi-Markov decision problems, recurrency conditions and algorithms
Authors: Federgruen A. , Tijms H. C.
Published: July 2016

This paper is concerned with the optimality equation for the average costs in a denumerable state semi-Markov decision model. It will be shown that under each of a number of recurrency conditions on the transition probability matrices associated with the stationary policies, the optimality equation has a bounded solution. This solution indeed yields a stationary policy which is optimal for a strong version of the average cost optimality criterion. Besides the existence of a bounded solution to the optimality equation, we will show that both the value-iteration method and the policy-iteration method can be used to determine such a solution. For the latter method we will prove that the average costs and the relative cost functions of the policies generated converge to a solution of the optimality equation.
The expected ultimate size of a carrier-borne epidemic
Authors: Lefèvre Claude
Published: July 2016

This paper is concerned with the expected ultimate size of the Downton carrier-borne epidemic (1968). The solution obtained is slightly recursive and it may be expressed as a power series in π, the proportion of the infected susceptibles becoming carriers. It generalizes the results of Dietz (1966) and Abakuks (1973) for some special cases of this model.
Superposition and decomposition of stationary point processes
Authors: Ito Yoshifusa
Published: July 2016

A recursion formula is obtained by rearranging Lawrance's (1973) result concerning the superposition of independent stationary point processes for which there exist joint probability density functions for the intervals between successive points. When these component point processes are identically distributed, the formula can in principle be inverted to describe their probability structure given that of the superposition process.
Extended block replacement policy with used items
Authors: Tango Toshiro
Published: July 2016

Block replacement policy, under which items are replaced at constant intervals of time and at failure, is wasteful because sometimes almost new items are also replaced. In order to eliminate this kind of waste, the following replacement policy is suggested.

(1) Items are exchanged for new items at time kT (k ··· 1, 2, …).

(2) If items fail in [(k – 1) T, kT – v), they are replaced by new items, and if in [kT – v, kT), they are replaced by used items.

The results of this ‘extended block replacement policy with used items' are studied for the Erlang lifetime distribution.
A cyclic queueing network with dependent exponential service times
Authors: Jacobs Patricia A.
Published: July 2016

A cyclic queueing network with two servers and a finite number of customers is studied. The service times for server 1 form an earma(1,1) process (exponential mixed autoregressive moving average process both of order 1) which is a sequence of positively correlated exponential random variables; the process in general is not Markovian. The service times for the other server are independent with a common exponential distribution. Limiting results for the number of customers in queue and the virtual waiting time at server 1 are obtained. Comparisons are made with the case of independent exponential service times for server 1.
Series expansions for the properties of a birth process of controlled variability
Authors: Cox D. R. , Isham V.
Published: July 2016

A birth process is studied in which the birth rate at any time is a function of the difference between the current population size and a target corresponding to unit growth rate. If this controlling function is a decreasing function of its argument a stabilizing effect is to be expected. By supposing that the controlling function varies very slowly, series expansions for the properties of the process are obtained, the leading term corresponding to a diffusion approximation. The sequence of births considered as a point process of controlled variability is examined and approximations to the interval distribution and covariance density obtained.
Optimal stopping in a semi-Markov shock model
Authors: Zuckerman Dror
Published: July 2016

We examine a failure model for a system existing in a random environment. The system accumulates damage through a shock process and the failure time depends on the accumulated damage in the system. The cumulative damage process is assumed to be a semi-Markov process. Upon failure the system must be replaced by a new identical one and a failure cost is incurred. If the system is replaced before failure, a smaller cost is incurred. We allow a controller to replace the system at any stopping time before failure time. We consider the problem of specifying a replacement rule which minimizes the total long-run average cost per unit time.
A continuous review (s, S) inventory system in a random environment
Authors: Feldman Richard M.
Published: July 2016

The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.
A unified approach to limit theorems for urn models
Authors: Holst Lars
Published: July 2016

An urn contains A balls of each of N colours. At random n balls are drawn in succession without replacement, with replacement or with replacement together with S new balls of the same colour. Let Xk
be the number of drawn balls having colour k, k = 1, …, N. For a given function f the characteristic function of the random variable ZM
= f(X
1)+ … + f(XM
), M ≦ N, is derived. A limit theorem for ZM
when M, N, n → ∞is proved by a general method. The theorem covers many special cases discussed separately in the literature. As applications of the theorem limit distributions are obtained for some occupancy problems and for dispersion statistics for the binomial, Poisson and negative-binomial distribution.
Replacement problem of a parallel system in random environment
Authors: Nakagawa T.
Published: July 2016

Råde considered a parallel system with n components in a random environment. The shocks cause the components to fail with certain probabilities. This paper considers the replacement policy for the above system in which it is exchanged before failure if the total number of failed components is more than k, and it is replaced if all of n components have failed. An optimum number of k which will minimize the expected cost is obtained. A numerical example is finally presented.
The area under the trajectory of carriers in a carrier-borne epidemic
Authors: Lefèvre Claude
Published: July 2016

This paper is concerned with the area under the trajectory of carriers in the Downton carrier-borne epidemic (1968). Its Laplace–Stieltjes transform is studied, and a recursive equation derived for it. Its mean is examined in some detail and the solution obtained is expressed as a power series in π, the proportion of the infected susceptibles becoming carriers.
A queuing-theoretic anomaly
Authors: Burke P. J.
Published: July 2016

The equilibrium delay distribution for a system with finite-source input, constant service time, and s > 1 servers, as determined by simulation, showed the existence of values of t such that the probability of delay greater than t is a non-monotone function of the input intensity. Since repeated efforts to remove this counter-intuitive result by debugging failed, a simple case was attacked by analytical and numerical methods; and the same anomaly was revealed. The method of analysis is described and an explanation of the result is offered.
A limit theorem for two-locus diffusion models in population genetics
Authors: Ethier S. N.
Published: July 2016

A limit theorem of Kurtz for perturbed operator semigroups is applied to show that the two-locus diffusion model in population genetics (allowing for selection, mutation, and migration) converges to a linkage-equilibrium diffusion model as Nc →∞, where N is the population size and c is the recombination fraction; in fact, with an appropriate change of variables, the limiting diffusion is what has been called the independent-loci diffusion model. This generalizes a result of Littler.
The joint density of the maximum and its location for a Wiener process with drift
Authors: Shepp L. A.
Published: July 2016

We give a simple expression for the joint probability density of: (a) the maximum value Y = max [X(t), 0 ≦ t ≦ T); (b) its location ; (c) the endpoint X(T), where X(t) = Xc
(t) is a Wiener process with drift, Xc
(t) = W(t) + ct, 0 ≦ t ≦ T. That is, we find the density p(θ, y, x) = p(θ, y, x; c, T) of , Y, X(T), p(θ, y, x; , Xc
(T) ∈ dx) is given by, 0 < θ < T, x ≦ y, 0 < y,
Some comments concerning a curious singularity
Authors: Feigin Paul D.
Published: July 2016

Consider the maximum likelihood estimation of θ based on continuous observation of the process X, which satisfies dXt
= θXtdt + dWt
. Feigin (1976) showed that, when suitably normalized, the maximum likelihood estimate is asymptotically normally distributed when the true value of θ ≠ 0. The claim that this asymptotic normality also holds for θ = 0 is shown to be false. The parallel discrete-time model is mentioned and the ramifications of these singularities to martingale central limit theory is discussed.
The stochastic geyser problem for sample quantiles
Authors: Book Stephen A.
Published: July 2016

Consider a sequence of observations Yk
= Xk
+ ek
, where {Xk
: 1 ≦ k < ∞} are i.i.d. random variables having distribution function F and {ek
: 1 ≦ k < ∞} are arbitrary random errors of observation. The stochastic geyser problem asks for conditions under which F can be uniquely determined from a knowledge of the sequence of Yk
's. The objective of the present article is to show that, if F is continuous and strictly increasing and the sample quantiles of successive blocks {Xn

+ 1
Xn

+ 2, …, Xn

+ K
} of particular lengths K can be a.s. estimated to within an error of size o(1) as K →∞, then we can almost surely determine F from a single realization of {Yk
: 1 ≦ k < ∞}.
The branching diffusion with immigration
Authors: Ivanoff B. G.
Published: July 2016

The branching diffusion with immigration is studied. Under general branching and diffusion laws, the process is shown to be mixing, according to Brillinger's definition. Brillinger's central limit theorem for spatially homogeneous mixing processes is generalized to prove that, under a renormalization transformation, the distribution of the branching diffusion with immigration converges to a completely random Gaussian random measure. In addition, the existence of a steady-state distribution is proven in the case of subcritical branching, and this distribution is shown to be mixing. Hence the steady-state random field also obeys a spatial central limit theorem.
On estimation of the integrals of certain functions of spectral density
Authors: Taniguchi Masanobu
Published: July 2016

Let g(x) be the spectral density of a Gaussian stationary process. Then, for each continuous function ψ (x) we shall give an estimator of whose asymptotic variance is O(n
–1), where Φ(·) is an appropriate known function. Also we shall investigate the asymptotic properties of its estimator.
Uniform conditional stochastic order
Authors: Whitt Ward
Published: July 2016

One probability measure is less than or equal to another in the sense of UCSO (uniform conditional stochastic order) if a standard form of stochastic order holds for each pair of conditional probability measures obtained by conditioning on appropriate subsets. UCSO can be applied to the comparison of lifetime distributions or the comparison of decisions under uncertainty when there may be reductions in the set of possible outcomes. When densities or probability mass functions exist on the real line, then the main version of UCSO is shown to be equivalent to the MLR (monotone likelihood ratio) property. UCSO is shown to be preserved by some standard probability operations and not by others.
Scheduling tasks with exponential service times on non-identical processors to minimize various cost functions
Authors: Weiss Gideon , Pinedo Michael
Published: July 2016

We consider preemptive scheduling of N tasks on m processors; processors have different speeds, tasks require amounts of work which are exponentially distributed, with different parameters. The policies of assigning at every moment the task with shortest (longest) expected processing time among those not yet completed to the fastest processor available, second shortest (longest) to the second fastest etc., are examined, and shown to minimize expected values of various cost functions. As special cases we obtain policies which minimize expected flowtime, expected makespan and expected lifetime of a series system with m component locations and N spares.
Correlation structure of teletraffic measurements from randomly-scanned calls arrival data
Authors: Lee P. A.
Published: July 2016

In teletraffic measurements, a call arrival process is commonly studied using a method with time-uniform or periodic scanning. The information recorded is the number of calls arrived between the scannings, from which data the number of scans between two successive calls is obtained. These later numbers are used as a measure of the interarrival times.

For an exponential call arrival process, except in the case of Poisson scanning, all other scanning schemes yield the number of scans which are not independent in any two interarrival intervals. By treating the problem as an interaction of two stationary stochastic point processes, we determine the exact joint probability distribution of the number of scans in two adjacent and non-adjacent interarrival intervals. An explicit expression for the correlation coefficient is also obtained.
Level crossings and stationary distributions for general dams
Authors: Rubinovitch Michael , Cohen J. W.
Published: July 2016

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.
On search for a Brownian target
Authors: Corwin Thomas L.
Published: July 2016

A target is assumed to move according to a Wiener process in ℝ1. The probability of detecting the target is computed in terms of the search effort which accumulates along the target's path. Under certain independence assumptions this probability is given by the expectation of an exponential functional of the process. It is shown in this note that the failure probability in a search for a Wiener target is asymptotically proportional to , where T is the accumulated time spent searching. The asymptotic failure probability is also shown to be independent of the position of the search in ℝ1. In a similar fashion, it is shown that the failure probability in a search for a Wiener target in ℝ2 is independent of the position of the search and asymptotically proportional to (c log T + l)–1, c >0
Convergence of successive approximations to a stochastic fixed point
Authors: Sanghvi Arun P.
Published: July 2016

This paper describes some sufficient conditions that ensure the convergence of successive random applications of a family of mappings {Γ
α
: α ∈ A} on a compact metric space (X, d) to a stochastic fixed point. The results are similar in spirit to a recent result of Yahav (1975).
Three characterizations of population strategy stability
Authors: Hines W. G. S.
Published: July 2016

In addition to the concept of the evolutionarily stable strategy (ESS), developed specifically for considering intraspecific conflicts, concepts such as the Nash equilibrium from game theory and the attractor or sink from dynamical systems theory appear relevant to the problem of characterizing populations of stable composition. The three concepts mentioned are discussed for one simple standard population model. It is found that evolutionarily stable strategies of one type are necessarily Nash equilibrium strategies, although the converse is not true. The dynamical systems characterization is found to provide a model for populations susceptible to invasion by ‘co-operative' strategies, but capable of evolving back in average to the original equilibrium.
Moments for a general branching process in a semi-Markovian environment
Authors: Whittaker Craig , Feldman Richard M.
Published: July 2016

A general branching process is extended to allow life length and reproduction probabilities to depend on randomly changing environmental states. First and second moments of the population size with respect to time are derived assuming that the environmental process is semi-Markovian. The results are similar to Markov-renewal type equations which allow, under discrete time, iterative computation of both moments of the population through time.
Evaluations of absorption probabilities for the Wiener process on large intervals
Authors: Park C. , Schuurmann F. J.
Published: July 2016

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦T

W(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.
Random walks on a dodecahedron
Authors: Letac G. , Takács L.
Published: July 2016

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn

+1 = j | Xn
= i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X
0 = A] and P[In = A | I
0 = A]. The methods used are applicable to other solids.
On the asymptotic geometrical behaviour of percolation processes
Authors: Schürger Klaus
Published: July 2016

In this paper the global behaviour of percolation processes on the d-dimensional square lattice is studied. Using techniques of Richardson (1973) we prove, under weak moment assumptions on the time coordinate distribution, the following result. There exists a norm N(·) on Rd
such that, for all 0 < ε < 1, we have that almost surely for all sufficiently large t the N-ball of radius (1 – ε)t is contained in η, (the set of all sites occupied by time t) and η, is contained in the N-ball of radius (1 + ε)t. Richardson (1973) derived the corresponding ‘in probability' result for a class of spread processes on Rd
, satisfying certain conditions.
The average distance between two convex sets
Authors: Groemer H.
Published: July 2016

In recent publications of Hadwiger, Bokowski, and Wills it has been shown that it is possible to evaluate the average distance between a fixed and a randomly distributed convex set in terms of the mean projection measures of these sets. In the present note it is pointed out that these and more general results can be obtained from a theorem concerning the relationship between integrals over arbitrary measure spaces and corresponding Lebesgue–Stieltjes integrals.
A note on the moments of the final size of the general epidemic model
Authors: Dunstan Ross
Published: July 2016

In the general epidemic model we study the first two moments of the final size. Beginning with the backwards equation, algebraic methods are used to find their asymptotic series expansions as the population size increases.
A note on the generalized Rao–Rubin condition and characterization of certain discrete distributions
Authors: Talwalker Sheela
Published: July 2016

Assuming that the original observation from a discrete distribution is subject to damage according to a binomial probability law, it is shown here that a class of discrete distributions consisting of binomial, Poisson and mixed Poisson distributions, is characterized by the generalized Rao–Rubin condition

0 <π, π′ < 1 are constants. Y denotes the resulting or observed random variable, taking the values r = 0, 1, 2, ….
Longest common subsequences of two random sequences
Authors: Chvátal Vacláv , Sankoff David
Published: July 2016

Given two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n ≦ 5, and f(n,2) where n ≦ 10. We study the limiting behaviour of n
–1
f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).
Weak convergence and first passage times
Authors: Gut Allan
Published: July 2016

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk > c‥kp
], c ≥ 0, 0 ≤ p < 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 < α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].
Generalized state-dependent Erlangian queues: Speculations about calculating measures of effectiveness
Authors: Conolly B. W.
Published: July 2016

The provision of easily calculable measures of effectiveness for generalised Erlang queues (state-dependent parameters λn
and μn
of arrival and of service) motivates speculation about the applicability of renewal theory. The application envisaged is justified by known results for certain models and its extension to an operationally more promising system is proposed. Use of the formula ‘L = λW' with ‘effective’ λ calculated by foregoing methods is likewise shown to be justified by known results for certain models and hence its wider applicability is conjectured. Mechanisms are discussed which may provide improved models, and investigation is made of choices of λn
and μn
which may lead to time dependent solutions having a prescribed form. The example of panic buying is considered as an example.
Maintainability of structures in Markov chain models under recruitment control
Authors: Davies G. S.
Published: July 2016

We consider a fixed size Markov chain model which suffers losses and admits controlled recruitment. The family of n-step maintainable structures is described geometrically and examined.
An extension of watanabe's theorem of characterization of poisson processes over the positive real half line
Authors: Bremaud P.
Published: July 2016

We give an elementary proof of the martingale characterization theorem for Poisson processes over the positive real half line. This theorem is due to Watanabe [8] in the case where the mean measure associated to the Poisson process is the Lebesgue measure.
The transition probabilities of the general stochastic epidemic model
Authors: Kryscio Richard J.
Published: July 2016

Recently, Billard (1973) derived a solution to the forward equations of the general stochastic model. This solution contains some recursively defined constants. In this paper we solve these forward equations along each of the paths the process can follow to absorption. A convenient method of combining the solutions for the different paths results in a simplified non-recursive expression for the transition probabilities of the process.
On Zipf's law
Authors: Woodroofe Michael , Hill Bruce
Published: July 2016

A Zipf's law is a probability distribution on the positive integers which decays algebraically. Such laws describe (approximately) a large class of phenomena. We formulate a model for such phenomena and, in terms of our model, give necessary and sufficient conditions for a Zipf's law to hold.
Conservative processes with stochastic rates
Authors: Saunders Roy
Published: July 2016

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.
Optimal reserves growth under a Wiener process
Authors: Foster W. F.
Published: July 2016

This paper considers a body whose funds accumulate according to a Wiener Process that has parameters which can be controlled at any stage. The process is bounded above by a level at which dividends (or savings) are set aside, and it is bounded below by a level at which a ‘rescue’ policy is invoked to avoid insolvency. Taking long-term dividend maximisation as the optimality criterion, first passage times are used to derive a general first order differential equation for the optimal control of the system at any reserves level, and this equation is solved fully for a certain class of problems. Examples are given of insurance and investment applications.
A weak convergence theorem for the empirical characteristic function
Authors: Kent John T.
Published: July 2016

The purpose of this paper is to show that the empirical characteristic function, when suitably normalised, converges weakly to a stationary Gaussian process whose autocovariance function is the theoretical characteristic function.
On the waiting-time and busy period distributions for a general birth-and-death queueing model
Authors: Natvig Bent
Published: July 2016

A general birth-and-death queueing model is considered with N waiting positions (0 ≦ N ≦ ∞), s servers (1 ≦ s ≦ ∞) and a first-come-first-served queueing discipline. The first and second order moments of the steady-state waiting-time (excluding service) for a non-lost arriving customer are given. By setting the busy period equal to the time where at least one service is in progress, we obtain the first and second order moments of the length of a busy period and also the distribution of the number served during it, given an arbitrary number of customers present originally. Using a direct approach all expressions are given in explicit forms which, although being far from elegant, are suitable for evaluation on a computer.
Dams with autoregressive inputs
Authors: Collings P. S.
Published: July 2016

In this paper a dam with autoregressive inputs and unit outputs is considered. Approximations for the distributions of the times to first emptiness and overflow are obtained using a Wald-type identity. The limiting behaviour of the dam content is also investigated
Networks of queues with customers of different types
Authors: Kelly F. P.
Published: July 2016

The behaviour in equilibrium of networks of queues in which customers may be of different types is studied. The type of a customer is allowed to influence his choice of path through the network and, under certain conditions, his service time distribution at each queue. The model assumed will usually cause each service time distribution to be of a form related to the negative exponential distribution.

Theorems 1 and 2 establish the equilibrium distribution for the basic model in the closed and open cases; in the open case the individual queues are independent in equilibrium. In Section 4 similar results are obtained for other models, models which include processes better described as networks of colonies or as networks of stacks. In Section 5 the effect of time reversal upon certain processes is used to obtain further information about the equilibrium behaviour of those processes.
On the tails of waiting-time distributions
Authors: Pakes A. G.
Published: July 2016

Results are given which relate the tail behaviour of the service and limiting waiting time distributions of a GI/G/1 queue. A limit theorem for the maxima of waiting times is given.
A note on the optimal control of carrier-borne epidemics
Authors: Wickwire K.
Published: July 2016

A model is proposed for the control of a deterministic carrier-borne epidemic by immunising susceptibles. An optimal control is of bang-bang type and it is proved that it can have at most one switch.
Asymptotic distribution for the coupon — collector's and sampling-tagging problems, when tagging affects catchability
Authors: Samuel-Cahn Ester
Published: July 2016

The coupon-collector's and sampling tagging problems are considered, under the model that each tagged element is γ > 0 times as likely to be caught at the next stage as every untagged element. Let WN
(γ, k) and SN
(γ, j) denote, respectively, the waiting time until k + 1 distinct elements are obtained, and the number of distinct elements in a sample of size j, when the population size is N. Complete characterisations are obtained for the limiting distributions of WN
(γ, k) and SN
(γ, j), in terms of the rates at which k and j tend to infinity with N.
Recurrence times of draft-patterns from reservoirs
Authors: Pegram G. G. S.
Published: July 2016

Expressions for the mean and variance of the recurrence time of non-overlapping draft-patterns of draft from a Moran Reservoir Model (discrete-state and discrete-time Markov chain) are derived using Feller's Renewal argument. In addition an expression for the mean recurrence time for self-overlapping patterns of draft is derived using run-theory.
A non-Markovian birth process with logarithmic growth
Authors: Grimmett G. R.
Published: July 2016

I show that the sumof independent random variables converges in distribution when suitably normalised, so long as theXksatisfy the following two conditions:μ(n)= E |Xn|is comparable withE|Sn| for largen,andXk/μ(k) converges in distribution. Also I consider the associated birth processX(t) = max{n:Sn≦t} when eachXkis positive, and I show that there exists a continuous increasing functionv(t) such thatfor some variableYwith specified distribution, and for almost allu. The functionv, satisfiesv(t) =A(1 +o(t)) logt. The Markovian birth process with parameters λn= λn, where 0 < λ < 1, is an example of such a process.
Convergence properties of age distributions
Authors: Keenay G. A. , Morgan R. W. , Ray K. H.
Published: July 2016

The literature on supply models for manpower planning in a hierarchy shows that an important consideration is the size of the discrepancy between the age distribution of the population and the stationary age distribution which would be reached if present policies were continued indefinitely. We establish the convergence properties of age distribution in various cases and show under what circumstances a convergent age distribution exists. In the convergent case we examine both the speed of convergence and the form of the stationary age distribution.
The candidate problem with unknown population size
Authors: Rasmussen Will T. , Robbins Herbert
Published: July 2016

The familiar problem of maximizing the probability of choosing the best from a group of N candidates, where N is known, is extended to the case of N unknown. An a priori distribution is assumed for N, and the case of a uniform distribution is examined. Let VN
denote the probability of choosing the best from a group of at most N candidates, then it is shown that lim
N→∞
VN
= 2e
–2.
Signalling over a Gaussian channel with feedback and autoregressive noise
Authors: Wolfowitz J.
Published: July 2016

We study in detail the case of first-order regression, but our results can be extended to the general regression in a straightforward manner. An average energy constraint ((1.2) below) is imposed on each signal. In Section 2 we give an optimal linear signalling scheme (definition and proof in Section 4) for this channel. We conjecture that this scheme is optimal among all signalling schemes. Then the capacity C of the channel is (see Section 5) – log b, where b is the unique positive root (in x) of the equation x
2 = (1 + g
2(1 + |α|x)2)–1. Here a is the regression coefficient, and g
2 is the ratio of the average energy per signal to the variance of the noise. An equivalent expression is C = ½log(1 + g2(1 + |α| b)2).
Radon-Nikodym derivatives, passages and maxima for a Gaussian process with particular covariance and mean
Authors: Bar-David Israel
Published: July 2016

We find expressions for the R–N derivative of the stationary Gaussian process with the particular covariance and mean, respectively, R(t, s) = max(1 – |t – s|, 0) and m(t)= aR(t, D), 0 ≦ D ≦ 1, within the time interval [0, 1]. We use these results, and a lemma on multiple reflections of the Wiener process, to find formulae for the probabilities of first passage time and maxima in [0, 1], and bounds on the former within [– 1, 1]. While previous work dealt extensively with the zero mean process, mean functions, as defined here, appear in signal detection and parameter estimation problems under the hypothesis that a rectangular signal centered at t = D is present in an observed process.