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De Bruijn's identity relates two important concepts in information theory: Fisher information and differential entropy. Unlike the common practice in the literature, in this paper we consider general additive non-Gaussian noise channels where more realistically, the input signal and additive noise are not independently distributed. It is shown that, for general dependent signal and noise, the first derivative of the differential entropy is directly related to the conditional mean estimate of the input. Then, by using Gaussian and Farlie–Gumbel–Morgenstern copulas, special versions of the result are given in the respective case of additive normally distributed noise. The previous result on independent Gaussian noise channels is included as a special case. Illustrative examples are also provided.

In this paper we study the asymptotic behaviour near extinction of (sub-)critical continuous-state branching processes. In particular, we establish an analogue of Khintchine's law of the iterated logarithm near extinction time for a continuous-state branching process whose branching mechanism satisfies a given condition.

The coalescent was introduced by Kingman (1982a), (1982b) and Tajima (1983) as a continuous-time Markov chain model describing the genealogical relationship among sampled genes from a panmictic population of a species. The random mating in a population is a strict condition and the genealogical structure of the population has a strong influence on the genetic variability and the evolution of the species. In this paper, starting from a discrete-time Markov chain model, we show the weak convergence to a continuous-time Markov chain, called the structured coalescent model, describing the genealogy of the sampled genes from whole population by means of passing the limit of the population size. Herbots (1997) proved the weak convergence to the structured coalescent on the condition of conservative migration and Wright–Fisher-type reproduction. We will give the proof on the condition of general migration rates and exchangeable reproduction.

We study in this paper a generalized coupon collector problem, which consists in analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the non-null coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one, which stochastically maximizes the time needed to collect a fixed number of distinct coupons. An computer science application shows the utility of these results., Comment: arXiv admin note: text overlap with arXiv:1402.5245

In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.

In this paper we connect Archimedean survival processes (ASPs) with the theory of Markov copulas. ASPs were introduced by Hoyle and Mengütürk (2013) to model the realized variance of two assets. We present some new properties of ASPs related to their dependency structure. We study weak and strong Markovian consistency properties of ASPs. An ASP is weak Markovian consistent, but generally not strong Markovian consistent. Our results contain necessary and sufficient conditions for an ASP to be strong Markovian consistent. These properties are closely related to the concept of Markov copulas, which is very useful in modelling different dependence phenomena. At the end we present possible applications.

In this paper, we investigate the functional central limit theorem for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional central limit theorem is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT., 10 pages

The purpose of this paper is twofold. On the one hand, we provide sufficient conditions for the excess wealth order. These conditions are based on properties of the quantile functions which are useful when the dispersive order does not hold. On the other hand, we study sufficient conditions for the comparison in the increasing convex order of spacings of generalized order statistics. These results will be combined to show how we can provide comparisons of quantities of interest in reliability and insurance.

In this paper we build the increasing convex (concave) order for the scalar product of random vectors with an upper (lower) tail permutation decreasing joint density. As applications, we revisit allocations of portfolio risks in financial engineering and of coverage limits and deductibles in insurance. Some related results in the literature are substantially updated.

A reflected Ornstein–Uhlenbeck process is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. It is an extended model of the traditional Ornstein–Uhlenbeck process being extensively used in finance as a one-factor short-term interest rate model. Under some mild conditions, this paper is devoted to the study of the analogue of the Cramer–Rao lower bound of a general class of parameter estimation of the unknown parameter in reflected Ornstein–Uhlenbeck processes.

In this paper we consider an M/M/c queue modified to allow both mass arrivals when the system is empty and the workload to be removed. Properties of queues which terminate when the server becomes idle are firstly developed. Recurrence properties, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no mass exodus. All of these results are then generalized to allow for the removal of the entire workload. In particular, we obtain the Laplace transformation of the transition probability for the absorptive M/M/c queue.

This paper presents a simple and computationally tractable method which recursively computes the stationary probabilities of the queue size in an M/G/1 queueing system with variable service rate. For each service two possible service types are available and the service rule is characterized by two switch-over levels. The computational approach discussed in this paper can be applied to a variety of queueing problems.

Silverman and Brown (1978) have derived Poisson limit theorems for certain sequences of symmetric statistics, based on a sample of independent identically distributed random variables. In this paper an incomplete version of these statistics is considered and a Poisson limit result shown to hold. The powers of some tests based on the incomplete statistic are investigated and the main results of the paper are used to simplify the derivations of the asymptotic distributions of some statistics previously published in the literature.

A recent paper in Advances in Applied Probability (Siegel (1976)) considered the duration of the time Tmn for a linear birth and death process to grow from a (large) initial size m to a larger size n. The main aim was to show that, when the birth rate exceeds the death rate, Tmn is close to its mean value, log n/m, with high probability. This paper establishes this result using much simpler techniques.

Using the theory of point processes, we give a formal treatment to queueing processes in the steady state. Based on this result, we obtain invariance relations between several quantities in GIG/c queues. As applications, the finiteness of their moments is discussed for G/G'/c queues. The basic results and notations used in this paper are contained in the author's previous paper (Miyazawa (1977)). FCFS MANY-SERVER QUEUES; STATIONARY INPUT; INVARIANCE RELATIONS; EXISTENCE OF MOMENTS; STEADY-STATE DISTRIBUTIONS; POINT PROCESS; PALM MEASURE

The chord length distributions of planar convex sets are discussed, particu- larly the density values at the extremes of the range; there is a qualitative distinction between polygons and sets with smooth boundaries. The distance between convex sets is related to the distance between distribution functions. This paper is concerned with recording some of the simpler properties of chord length distributions (henceforth abbreviated to c.l.d.) of compact con- vex figures in R2. Detailed calculation of c.l.d.'s are exercises in integration. In the case of polygons the results can be expressed in terms of elementary functions, but are usually highly tedious, requiring different expressions over different ranges. In this paper Kwill denote a compact convex set in R2, containing more than one point; Jr will denote the set of such sets and ((.) the set of equivalence classes of.J under the Euclidean group of motions, M2. The measure on the set of lines in R2 will be the M2 invariant measure, u. If a line G meets K we may write G t K and the set of incident lines as G(K), with S(G) the chord length. If an origin is chosen in K a line can be parametrised by its perpendicular distance, p, and the orientation p of that perpendicular. It is well known (Santal6 (13), p. 28) that the invariant line measure is dG = dpd4, so that, in this parametrisation, ,L is identified with Lebesgue measure, u2, in R2. It is shown in Sulanke (14) that S is a measurable function on G(K). As #u(G(K)) is equal to the perimeter, L, of K the distribution function of chord length is

We assume that the number of claims occur according to a renewal process and treat the number of claims that occur and are reported in a certain time interval as a renewal process with random displacements. We obtain a renewal equation for the mean value function and an integral equation for the Laplace transform of the distribution of the claims that are reported. We also give asymptotic expressions for the mean value function and calculate the generating function in the case where the renewal process is a Poisson process. This matter is a part of the IBNR-problem in insurance mathematics. INCURRED BUT NOT REPORTED CLAIMS; RENEWAL PROCESS WITH RANDOM DISPLACEMENTS ntroduction At a given point of time one has only partial knowledge of an insurance business. For instance, there is always a time lag between the occurrence of a road accident and the reporting of and full knowledge of the claim amount due to that accident. This lack of information generates many questions. One, which this paper touches, is: if we observe M(t) occurred and reported claims during the interval (0, t), how many claims occurred in that period? On the accounting date one has to answer this question and make an estimate of the number of claims which occurred during the last year. The starting point will be assumptions of the number of claims that occur in (0, t). Since the days of Filip Lundberg one has assumed that this is described by a Poisson process. Several authors have tried to generalize Lundberg's model. Without intending to make a complete list we mention Ove Lundberg, Sparre Andersen and Jan Grandell. We will pursue the course of Sparre Andersen who in 1957 presented a paper [1] where the epochs of claims form a renewal process. As to the distribution of the time lag between the occurrence of a claim and the report of that claim there are several possible ways. We will make the assumption that the lags are independent and equally distributed, which seems to be a reasonable assumption as a first approximation at least when the claim amounts are of a fairly equal size. Received in revised form 21 March 1973. 382 This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms A stochastic modelfor time lag in reporting of claims 383 A model We assume that Feller [3], ch. XI is known. Assume that claims occur according to a delayed renewal process on [0, oo). Number the claims 0, 1, 2, -.. in the order in which they occur. So is a stochastic variable representing the time between the starting point zero and the occurrence of the first claim and So has the distribution function Fo. T, is a stochastic variable representing the time between the occurrence of claim number n 1 and claim number n, n = 1,2, ---. The variables TI, T2, -" have the same non-arithmetic distribution function F, with F(O) = 0. Put P, = E T, and aF = var T. and assume that ao 1, This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms 384 JAN-ERIK KARLSSON A fundamental property of the renewal process is that N(t) > n + 1 if and only if S, n if and only if S, + X, . We omit the index Fo, if P(So = 0) = 1. Thus we have MFo(t) = I e,(t) 0 and VFo(t) = P((t) P((t) = 1) = (G*Fo*F*)(t) 0 0 or putting Go = G,*Fo VF,(t) = (Go * U) (t). Thus VFo(t) is the solution of the renewal equation VFo(t) = Go(t) + VFr(tx)F {dx}. Conditioning by So, we see that VFo(t) = (V* Fo)(t). We can also get the renewal equation for VFo(t) by the usual renewal reasoning: EMFo(t) = f E(MFo(t) lSo + X0 = S, So + T1 = y)Pso+xo{ds} Pso+r,{dy} where (0 if s > t, y > t, E(MF(t) So + Xo = s, S + T1 = ) = 1 if s t, y> t, EM(t y) if s > t, y t, 1 +EM(t y) if s ?t, y t. This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms A stochastic modelfor time lag in reporting of claims 385 This technique will be used later. According to the renewal theorem for a delayed renewal process, we have for h > 0 lim (VFo(t + h) VFo(t)) = h I/,. If we put z(t) = VFo(t) t /ip it can be shown that Z(t) satisfies Z(t) = z(t) + f (t x) F{dx} with z(t) = (1 F(y))dy (1 Go(t)). Since z(t) is directly integrable (we assume that Po = ESo All use subject to http://about.jstor.org/terms A stochastic model for time lag in reporting of claims 387 References [1] ANDERSEN, E. SPARRE (1957) On the collective theory risk in the case of contagion between claims. Transactions XVth International Congress of Actuaries, New York. Vol. II. [2] DALEY, D. J. AND VERE-JONES, D. (1972) A summary of the theory of point processes. Stochastic Point Processes: Statistical Analysis, Theory and Applications (P. A. W. Lewis, ed.), Wiley, New York. 299-383. [3] FELLER, W. (1970) An Introduction to Probability Theory and its Applications. Vol. 2, 2nd edition. Wiley, New York. [4] PRABHU, N. U. (1965) Queues and Inventories. Wiley, New York. [51 THEDtEN, T. (1967) On point .v.vtoms in R, under random motion. Doctor's thesis, Royal Institute of Technology, Stockholm. This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ T W(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ T W(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.

This paper investigates a system whose deterioration is expressed as a continuous-time Markov process. It is assumed that the state of the system cannot be identified without inspection. This paper derives an optimal policy minimizing the expected total long-run average cost per unit time. It gives the optimal time interval between successive inspections and determines the states at which the system is to be replaced. Furthermore, under some reasonable assumptions reflecting the practical meaning of the deterioration, it is shown that the optimal policy has monotonic properties. A control limit rule holds for replacement, and the time interval between successive inspections decreases as the degree of deterioration increases.

Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple The original purpose of this paper was to find a workable expression for the transform of the Brownian excursion area. We wanted to use two well-known results: Kac's formula for Brownian functionals and Levy's local time decompos- ition. We actually found that these two powerful tools lead also to alternative and sometimes simpler proofs for other results on Brownian excursion prob- abilities: maximum and local time distributions. We were also able to derive a useful general result on Brownian excursion symmetric additive functionals. Finally, we also wanted to understand some curious relations that had been observed by earlier authors: connections between hitting-time densities and maximum distributions. Using simple probabilistic arguments and a formula for Jacobi's third 0 function, we can explain these connections in direct terms. The paper is organized as follows. In Section 1, we summarize basic notations and known results. Sections 2 and 3 are short surveys of Kac's and Levy's results. These tools are applied in Sections 4 and 5 to Brownian excursion maximum and local time. Section 6 contains the main results of the paper: a useful new form for the transform of the Brownian excursion area density and a simple relation for functionals based on a symmetric positive function. Section 7 explains those curious relations alluded to earlier between Brownian excursion probabilities. 1. Basic notations and known results

Over a finite interval, a Gaussian stationary process can be approximated by a finite trigonometric sum, and the error introduced by the approximation can be exactly bounded, as far as the distribution of the upper tail of the maximum is concerned. A simple case is exhibited, where a narrow band process is well approximated by means of a two-term trigonometric representation. GAUSSIAN PROCESS; DISTRIBUTION OF MAXIMUM; NARROW-BAND PROCESSES; FOURIER SERIES Introduction The motivation for the work described in this paper arose from dissatisfaction with the state of the theory of the distribution of the maximum of a Gaussian stationary process with smooth sample functions over a finite interval. It is well known that an explicit solution to this problem can be obtained only in very few cases. In a recent paper, Shepp and Slepian (1976) state that to the best of their knowledge it is known for four non-trivial covariance functions only. The literature abounds with asymptotic theorems for the distribution of the maximum above high levels. For a survey of results and bibliography, see Blake and Lindsey (1973) and Vanmarcke (1975). Very little is known, however, about the precision of the asymptotic approximations when they are used for finite levels. These asymptotic approximations turn out in fact to be numerically unsatisfactory at these finite levels when compared with simulation results. In this paper, it is shown that, over a finite interval (T, + T), a Gaussian stationary process can be approximated by a finite trigonometric sum, and that the error introduced by the approximation can be exactly bounded, as far as the distribution of the upper tail of the maximum is concerned. A simple case is exhibited where a narrow-band process is well approximated by means of a two-term trigonometric representation. @ Applied Probability Trust 1982 0021-9002/82/050333-11 $01.35

In previous papers (1976), (1977) limit theorems were obtained for the classical Ising model in the absence of an external magnetic field, thereby providing a basis for asymptotic inference. The present paper extends these results to arbitrary external magnetic fields. Statistical inference for this model is important because its nearest-neighbour interactions provide a natural first approximation to spatial interaction among binary variables located on square lattices. The most interesting behaviour occurs in zero field and at or beyond the critical point. In this case, the central limit result for nearest-neighbour interactions requires an unusual norming, the limiting variances may depend on the nature of the boundary conditions, and there cannot be any central limit result for external magnetic field. The implications of these phenomena for statistical inference are also discussed. In particular, the maximum likelihood estimator of magnetic field is not consistent. Rather it appears to have a non-trivial asymptotic distribution.

Sufficient conditions for strong ergodicity of discrete-time non-homogeneous Markov chains have been given in several papers. Conditions have been given using the left eigenvectors ψ n of Pn (ψ nPn = ψ n ) and also using the limiting behavior of Pn. In this paper we consider the analogous results in the case of continuous-time Markov chains where one uses the intensity matrices Q(t) instead of P(s, t). A bound on the rate of convergence of certain strongly ergodic chains is also given.

This paper continues the author's study of age-dependent branching processes allowing immigration. In this paper the multidimensional case is considered. A sufficient condition is obtained for the existence of a legitimate limiting distribution. Several corollaries are obtained, which generalize many of the results of the discrete theory and those of the one-dimensional continuous time model.

This paper is concerned with nearest-neighbour systems on the coloured lattice (unordered state space). It extends the paper of Strauss (1975) on clustering in the two-colour case. Comparison is made with the more general methods of Besag (1974). Some tests are developed, and illustrated with an example. NEAREST-NEIGHBOUR SYSTEM; MARKOV RANDOM FIELD; CLUSTERING; QUALITATIVE DATA

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived. MARKED BIRTH-AND-DEATH PROCESS; MULTIVARIATE LAGRANGE DIFFERENTIAL EQUATIONS; MULTITYPE BRANCHING PROCESS Introduction In a recent paper Williams (1969) has considered the following process. We have a population of elements developing as a linear birth-and-death process and each element bears a number of inanimate marks. When a marked element dies then it and its marks are removed from the population; when a marked element divides, its marks segregate independently and without loss, each with probability I into one of the two daughter elements. It is assumed that marking an element does not affect its birth and death rates (cf. Clifford and Sudbury (1972)). Williams' process was proposed as a model of an experiment conducted on bacteria (the elements) infected by phage (the inanimate marks), by Meynell (1959). (The bacteria are lysogenic with respect to the phage, which means that the phage do not interfere with the behaviour of the bacteria.) His object was to use the marking process in order to separate the birth and death rates of the bacteria; if we denote birth and death rates by A and p respectively, then the mean-value measures on the bacterial population (assumed to develop as a linear birth-and-death process) will estimate the difference (A p). As the marks are inanimate, however, they will only be affected by p and so mean-value measures on the mark population enable us to estimate p (see Williams (1969)). The papers of Meynell and Williams taken together provide a most interesting example of a virological experiment and the probability model of that experiment. Received in revised form 10 August 1973 Research sponsored by a Science Research Council grant and a World Health Organisation research traineeship. 423 This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms 424 BYRON J. T. MORGAN Williams considered the mean-value solution of his model; this was adequate for Meynell's experiment in which population sizes were large. Here we shall give a fuller probabilistic statement of the model and make some contributions to its solution. Two differences from Williams' work lie in the adoption of an upper bound, s, to the number of marks any element can have, and the assumption (except where stated) of birth and death rates independent of time. We 'shall assume general initial conditions which, in Meynell's experiment, correspond to the distribution of phage over bacteria once these have been initially exposed to a phage population. Williams took this distribution to be Poisson; the distribution of phage over bacteria has, however, been the subject of much probabilistic study (e.g., see Gani (1965) and Morgan (1971)). Finally, let us note that the value s = I, on which we shall particularly concentrate, probably gives the most realistic description of Meynell's experiment. 2. The probability model The model is a continuous-time Markov process with discrete states. Let N,(t) be a random variable denoting the number of elements with i marks at time t, i = 0, 1,2, ..., s. Let n = (n0,n1, l , n.., ), Iu = (Uo, u1, *,us), N(t) = (No(t), N(t), --,N,(t)), and suppose N,(O) = ai, i = 0, 1,2, --, s. Let P(n; t) = P(N(t) = n), then corresponding to death we have the transitions, (No, ..., Ni, ---, N,) (No, --, Ni 1, , N,) at probability rate pNi, i = 0, 1,2, ,s, while corresponding to birth we have the following transitions (No, N,) ,(No + 1, .., N,) at probability rate, 2 =1 N 2-' +,ANo, (No, , Ni ,N2i,..., X (No, Ni + 2, , N2iI, at probability rate, This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms On the distribution of inanimate marks over a linear birth-and-death process 425 where we use [is] to denote the integral part of is, and (No, " , Ni, * * *, Nk i Nk, "', Ns) -*(No, ..., Nj + 1, ..., Nk-i + 1, ..., Nj 1, ..., Nl) at probability rate 2N 2-k' i = 1,2,...,C[(k 1)], k = 3,4,...,s (smaller k give transitions already considered above). It is readily verified that the sum of the transition rates is (A + yp) X = oN,, as required in the overall birth-and-death process. Introducing the operator Di, which increases the argument ni of a function of n by 1, we then deduce that the Kolmogorov forward equations (for all possible n) are, d S S (1) +(n; t) P(n; t)(A+ t)1 (n2 + D,P(n;t)(n +1) dt ==0 1=0

We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in theM/GI/squeue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived forM/GI/s/kwith finite and infinitek.Similar approximations are possible forM/GI/s/k(k< +∞).

This paper describes an extension of the SDM scheme introduced by Priestley (1980) to the problem of the identification of non-linear systems using only input/output records. It is shown that, under quite general assumptions on the structure of the system, it is possible to identify a non-linear relationship between the input and output processes via a modified version of the Kalman-type algorithm previously applied to the study of non-linear time series models. The paper includes a numerical study of data generated from various types of non-linear systems. NON-LINEAR MODELS; KALMAN FILTERING; NON-PARAMETRIC FUNCTION FIT'ING

We adapt arguments from a paper of Seymour and Welsh concerning percolation probabilities on the infinite square lattice L to show that for certain regions R in L, if there is a positive probability of having an infinite open path in L starting at the origin then there is also a positive probability of having such a path within R. PERCOLATION; CRITICAL PROBABILITY; SQUARE LATTICE This note is based on the paper of Seymour and Welsh (1978) and we adopt notation from that paper. We denote by L the infinite square lattice, that is the set of points (i,j) in the plane with integer coordinates, together with edges joining each point (i, j) to its four nearest neighbours (i + 1, j), (i 1, j), (i, j + 1), (i, j 1). Fix some real number p with 0 p 1, and suppose that each edge of L has probability p of being open and probability q = 1 p of being closed, independently of all other edges. Regard the origin as a source of fluid and suppose that the fluid can flow along open edges but not along closed edges. We say that a point v of L is wet if there is an open path (that is, a path consisting of open edges) from the origin to v, and otherwise v is dry. We may enquire which values of p give us a positive probability of having an infinite number of wet vertices. Note that by K6nig's lemma, for any assignment of open or closed to each edge of L, there is an infinite number of wet vertices if and only if there is an infinite open (self-avoiding) path in L starting at the origin. Given any non-negative function f defined on the non-negative integers with f(0) = 0 let R(f) denote the region of L in the first quadrant above the function f, that is R (f) denotes the subgraph of L induced by those points (i, j) in L with i -0 and jf(i). Theorem. Fix p and suppose that for some integer k f(n)exp{-(logn)k}--O as n -oo. Received 27 November 1978; revision received 15 January 1979. * Postal address: London School of Economics and Political Science, Houghton St., London WC 2A 2AE, U.K. Research partially supported by NRC grant A9211.

This paper deals with the existence of solutions of a stochastic integral equation of the Volterra type and their asymptotic behaviour. Investigations of this paper use the concept of a measure of non-compactness in Banach space and fixed-point theorem of Darbo type. An application to a stochastic model for chemotherapy is also presented.

Many species of birds have a characteristic clutch size which is either fixed at k or is of the form k or k + 1 for some appropriate integer k. 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 time t for each finite t. 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.

Various univariate classes of life distributions have been introduced in the mathematical theory of reliability. Recently there has been much interest in obtaining multivariate versions of these classes. Although there have been many different approaches, this document mentions only two because of their nice closure properties: the multivariate IFRA class of Block and Savits (1980) and the multivariate NBU class of Marshall and Shaked (1982). Both of these classes are closed under deletion, conjunction, convolution and weak limits. This paper introduces a multivariate IFR class that has similiar closure properties. its spirit also closely parallels that found in the two previously mentioned papers. Other definitions of a multivariate IFR class have been proposed by Marshall (1975), but none possess all desirable closure properties. Section 2 presents some preliminary facts about log concave functions. Section 3 contains new characterization of the (univariate) IFR class. The multivariate genralization and properties thereof are given in Section 4. A useful alternative condition is delineated in Section 5. Finally, this class is compared with the multivariate IFRA class of Block and Savits (1980) in Section 6.

This paper considers an infinite dam fed by a discrete input Xt during the time interval [t, t + 1), t = 0, 1, 2, ···. At time t – 0 there is an output Yt = min(Zt –1, + Xt –1, r) from the dam leaving behind the amount Zt = max(0, Zt –1, + Xt –1, r). The probability Pr(Zt = i), i = 0, 1, 2, ··· is discussed under the strict assumption that r > 1 and the given initial condition that Z 0 = u, u = 1, 2, ···. The generating function technique has been used throughout the paper.

This paper considers the generalized likelihood ratio (GLR) test or its modification dealing with nested models. The algorithm for evaluating the critical values and the error-rates for the canonical tests are provided; a table of critical values of a class of GLR tests is also given. The test proposed in the paper has applications in time-series model selection. This paper explores a generalization of the likelihood-ratio test in order to deal with testing a hypothesis against nested alternatives. Though such a testing situation is very common in the practice of time-series analysis (take for instance testing the hypothesis of white noise against the autoregressive hypotheses of various orders), there does not seem to be an established general method of dealing with it, except probably for the one which chooses the most complicated as the alternative, which is not necessarily a reasonable choice when a simpler alternative is true. The paper proposes a simultaneous use of several likelihood-ratio statistics which is termed a generalized likelihood ratio (GLR) test and a use of a transformation method of such statistics in cases where an algorithmic impasse with respect to the evaluation of simultaneous distribu- tion arises.

In a previous paper, the author has described a method for obtaining envelope processes for oscillatory stochastic processes. These are processes which can be represented as the output of a time-varying linear filter whose input is a stationary process. It is shown in this paper that the proposed definition of the envelope process may not be unique, but may depend on the particular representation of the oscillatory process chosen. It is then shown that for a class of oscillatory processes which is of particular interest, the class of transient processes, there is a class of natural representations which all lead to a unique envelope process.

This paper discusses infinite server queues whose input is a Phase Type Renewal Process. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations are provided with as many as ten phases. The paper contains some useful explicit formulas and also discusses the interesting special case where the service time is also of phase type. The Phase Type Distributions include a wide variety of models such as generalized Erlang, hyperexponential (mixtures of a finite number of exponentials) as very special cases and possess great versatality in modeling a number of interesting qualitative features such as bimodality.

Foster's criterion for positive recurrence of irreducible countable space Markov chains is one of the oldest tools in applied probability theory. In various papers in JAP and AAP it has been shown that, under extensions of irreducibility such as ϕ -irreducibility, analogues of and generalizations of Foster's criterion give conditions for the existence of an invariant measure π for general space chains, and for π to have a finite f-moment ∫π (dy)f(y), where f is a general function. In the case f ≡ 1 these cover the question of finiteness of π itself.In this paper we show that the same conditions imply the same conclusions without any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically. The proofs involve detailed consideration of the structure of the minimal subinvariant measures of the chain.The results are applied to random coefficient autoregressive processes in order to illustrate the need to remove irreducibility conditions if possible.

The matrix-geometric results of M. Neuts are extended to ergodic row-continuous bivariate Markov processes [J(t), N(t)] on state space B = {(j, n)} for which: (a) there is a boundary level N for N(t) associated with finite buffer capacity; (b) transition rates to adjacent rows and columns are independent of row level n in the interior of B. Such processes are of interest in the modelling of queue-length for voice-data transmission in communication systems. One finds that the ergodic distribution consists of two decaying components of matrix-geometric form, the second induced by the finite buffer capacity. The results are obtained via Green's function methods and compensation. Passage-time distributions for the two boundary problems are also made available algorithmically. MATRIX-GEOMETRIC; GREEN'S FUNCTION; COMPENSATION In a previous paper [10] a theoretical treatment of finite row-continuous bivariate Markov chains B(t)= [J(t), N(t)] was developed, providing an algorithmic basis for finding their ergodic distributions and associated passagetime moments. The continuous-time chain B(t) with state space = {(j, n):'0 j J, 0 n N} was described as row-continuous in the sense that the marginal process N(t), indexed by row coordinate n, changed at transition epochs by at most 1. In the present paper we restrict our discussion to those row-continuous chains for which the transition rate matrices, v°, vn, v, describing rates local to row n, are independent of n for each 1 _ n 5 N 1. For n = 0, one has vo = 0, and for n = N, v = 0. Such processes may be described as row-homogeneous, row-continuous processes modified by two retaining boundaries, as for earlier similar univariate contexts [4]. This research was conducted at the MIT Laboratory for Information and Decision Systems with partial support provided by the USAF OSR Grant Number AFOSR-79-0043, U.S. Air Force Geophysics Laboratory Grant Number F19628-80-C-0003, and the National Science Foundation Grant Number NSF/ECS 79-19880. © Applied Probability Trust 1988

In previous papers (1976), (1977a), (1979) limit theorems were obtained for the classical Ising model, and these provided the basis for asymptotic inference. The present paper extends these results to more general Ising models. In two and more dimensions, likelihood inference for the thermodynamic parameters (i.e. the interaction energies) is effectively impossible. The problem is that the error in locating critical and/or confidence regions is as large as their diameters. To remedy this requires more accurate characterizations of the partition functions, but these seem unlikely to be forthcoming. Besag's coding estimators for these parameters are inverse hyperbolic tangents of the roots of simultaneous polynomial equations and hence avoid such location errors. However, little is yet known about their sampling characteristics. Finally, likelihood inference for lattice averages (an alternative parametrization) is straightforward from the limit theorems.

This paper aims to demonstrate that the general Crump–Mode–Jagers branching process may be used in a natural way to model the asymmetric growth of budding yeast cells. The models obtained are generalisations of the deterministic model proposed by Hartwell and Unger (1977): all the results that are derived in that paper may be obtained using branching-process methods, but such methods also apply when account is taken of the biologically obvious fact that the various phases of the cell growth are of random rather than fixed duration. In their full generality, branching processes involve more parameters than can be estimated by experiment, but we present below a special case in which this problem is not likely to arise. A recent paper, Lord and Wheals (1980), discusses more of the biological background than is appropriate here. In the present paper, we show how certain statistical procedures for our model may be developed.

We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in the M/GI/s queue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived for M/GI/s/k with finite and infinite k. Similar approximations are possible for M/GI/s/k (k < +∞).

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index. Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions. The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4]. In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γ n (p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n (s) the number of vertices in the union of all those components of Γ n (p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n (s) and the limit distribution of ρ n (s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

This paper is concerned with the characterization of the cumulative pensionable service over an individual's working life that is made up of random lengths of service in different employments in a given industry, under partial coverage, transferability, and a uniform vesting rule. This characterization uses some results that are developed in the paper involving a functional and cumulative constrained sojourn times (constrained in the sense that if a sojourn time is less than a given constant it is not counted) in semi-Markov processes.

In a previous paper, Natvig [4], we suggested a new measure of the importance of a component in a coherent system and derived some of its properties. The measure is for the case of components not undergoing repair proportional to the expected reduction in remaining system lifetime due to the failure of the component. In the present paper, we arrive at the whole distribution of this reduction in remaining system lifetime. Furthermore, for the case where components have proportional hazards, and are not repaired, a speculation of another measure is given. This measure is proportional to the derivative of the expected total lifetime of a new system with respect to the inverse of the component's proportional hazard rate.

In a previous paper a branching Poisson process model was derived to explain deviations from a Poisson process in computer failure patterns. Physically the deviations arise because an attempt to repair a computer is not always successful and the failure recurs a relatively short time later. In this paper we discuss the implications of this model for the use and maintenance of computers.

This paper treats the well known model of sib mating of individuals with autosomal genes; a single locus is studied. The general derivations are valid for any number of alleles. At the end of the paper, some simple examples are included; here several familiar results concerning models for two alleles are obtained as special cases of the general theory.

M(x) = max[kj Mk _ x]. M(x) + 1 is then the first passage time out of the interval ( oo, x] for the random walk process S,. In this paper we shall concern ourselves with just those cases in which M(x) is a proper random variable with EM(x) < co and, when suitably normed, possesses a limit distribution as x -) co. It will be shown that M(x) can possess such a limit if and only if the random variables Xi belong to the domain of attraction of one of a certain group of stable laws and the limit law will be obtained under these circumstances. This result (Theorem 2) constitutes a generalization from the case of non-negative summands of various limit results of classical renewal theory (see for example Feller [6], 359-360). The significance of this type of generalization has previously been explored in Heyde [8]; Theorem 2 of this paper is in fact an extension of Theorem 4 of [8]. In order to obtain the limit distribution mentioned above we need a version of