Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn) of Mn, define Note that v−, v+ depend only on C1, C2, …, Cn− 1; Vm only on cn; We shall as notational convenience dictates and as will be apparent from the context regard v±n as functions on Mn− 1 or on higher order moment spaces and also regard Vn as a function on moment spaces of order higher than n.

Let U denote the set of all integers, and suppose that Y = {Yu; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au; u ∈ U} be a sequence of real numbers with Σuau2 = 1. Then Xu = ΣwawYu–w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu) = 0, E(Xu2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if maxu |au| is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy ≦ g maxu |au | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w1, … wn), (Xw1, … Xwn) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filter a is studied.

A model for the service time structure in the single server queue is given embodying correlations between contiguous and near-contiguous service times. A number of results are derived in the case of Poisson arrivals both for equilibrium and the transient state. In particular, Kendall's (equilibrium) result P (a departure leaves the queue empty) = 1 — (mean service time)/(mean inter-arrival time) is found still to hold good. The effect of the correlation on the mean and variance of the equilibrium queue length distribution is examined in a simple case.

This survey article, most of whose results are described in greater detail in Munk and Cartwright (1966), which will hereafter be abbreviated to MC, describes methods which aim to separate the response of the sea level at a given place due to various exciting forces such as gravity, solar radiation, non-linear effects, and weather. In so doing, it provides predictors for sea level which are formally simpler and somewhat more accurate than those given by the classical methods.

One of the troublesome problems in the extension of queueing theory to more complex systems of servers is the development of models suitable for the analysis of assembly lines and similar structures. Results in special cases are given in [2], [5], and [6]. A reasonable model would assume:

This paper deals with the problem of predicting or extracting a certain type of non-stationary sequence or signal from another sequence composed of the signal and a stationary noise. The noise is assumed to be uncorrelated with the signal. Although all sequences in the body of this paper are multivariate, for purposes of discussion, this introduction will deal with only the univariate case.

It is well known (Lee 1960), especially to communications and control systems engineers, that the weighting function (unit impulse function) of a linear causal system may be obtained by cross correlating its output with its input. In general a Wiener-Hopf integral equation must be solved for the weighting function; but if the input were “white noise” of unit spectral density then, as shown by Lee (1950), the weighting function would be equal to this cross correlation function for positive values of its argument. Although a perfectly “white” input cannot be obtained in practice, a number of modifications of this technique have been used; Anderson, Buland and Cooper (1959), for example, describe the use of specially selected samples of “discrete-interval binary noise” in a problem in adaptive control.

This paper is a sequel to [1] and considers a more realistic formulation of the same question: that of finding an optimal policy for controlling the path of a space-ship as it moves towards its target. The difference here is that we no longer suppose there is an infinite quantity of fuel, always available at a fixed price, for modifying the current direction of motion. This complicates the problem of reducing the final miss distance, by introducing an extra variable. As before, we shall be particularly concerned to find a control procedure which always minimizes the mean square terminal miss. From the theoretical point of view we are also interested to see whether the techniques used to approximate the optimal policy can be extended, and how far we shall be forced to adopt a new approach. Results are derived which provide bounds on the form of the optimal policy. The derivation depends on a comparison technique whose validity is intuitively obvious, but which is still only a conjecture. However, further confirmation is obtained in the quadratic case from asymptotic expansions giving the form of the solution both when the space-ship is far away from its target and during its final approach.

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process {vn, n = 1,2, …}, recursively defined by K is a positive constant, τ1, τ2, …; Σ1, Σ2, …; are independent, non-negative variables, with τ1, τ2,…, identically distributed, similarly, the variables Σ1, Σ2, …, are identically distributed.

Let Y1, Y2, … be a sequence of independent and identically distributed Poisson random variables with parameter λ. Let Sn = Y1 + … + Yn, n = 1,2,…, S0 = 0. The event Sn = n is a recurrent event in the sense that successive waiting times between recurrences form a sequence of independent and identically distributed random variables. Specifically, the waiting time probabilities are (Alternately, the fn can be described as the probabilities for first return to the origin of the random walk whose successive steps are Y1 − 1, Y2 − 1, ….)

A single server queueing system with Poisson input is considered. There are a finite number of types of customers and the service time of the nth customers depends on the types of the nth and the (n – l)th customers. The time dependence of the queue size process will be studied, (it will be clear how the methods of the paper can be applied to other processes of interest,) and limiting as well as transient results will be given.

Let R be a bounded open subset of N-dimensional Euclidean space EN,N ≧ 1, let {xt: t ≧ 0} be a separable Brownian motion starting at a point x ɛ R, and let τ = τR be the first time the motion hits the complement of R. It is known [1] that if g is a bounded measurable function on the boundary ∂R of R, then h(x) = Ex[g(xτ)] is a harmonic function of x ɛ R which “solves” the Dirichlet problem for the boundary function g; i.e., Δh = 0 on R, where Δ is the Laplacian. In elastic plate problems, one must solve the biharmonic equation subject to certain boundary conditions. For the more important applications, these boundary conditions involve the values of u and the normal derivative of u at points of ∂R. Even though a treatment of this Neumann type problem is not available at this time, some things can be said about biharmonic functions and their relationship to Brownian motion. We will show, in fact, that u(x)= Ex[τ(xτ)] is a biharmonic function on R which “satisfies” the boundary conditions (i) u=0 on ∂R and (ii) Δu= −2g on ∂R, provided g satisfies certain hypotheses. More generally, we will show that u(x)=Ex[Δkg(XΔ)] is polyharmonic of order k + 1 on R (i.e., Δk + 1u = Δ(Δku) = 0 on R) and that it satisfies certain boundary conditions. A treatment of the special case g ≡ 1 on ∂R can be found in [3].

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.

This note presents a new “uncertainty function” of finite probability distributions that arises in sequential search theory. The function arises as the finite term in an asymptotic expression for an expected search time as the allowable “error probability” approaches zero. This function is not a function of entropy; it may be of more relevance than entropy to workers in certain fields for describing the complexity of a finite distribution.

The purpose of this note is to point out some difficulties which arise in connection with the stochastic calculus used in the previous paper and to suggest some ways in which these difficulties may be overcome. It turns out that if one wishes to restrict the application of the stochastic calculus to stochastic processes whose sample functions are step-functions with probability one, then difficulties of the following sort arise. Suppose X(ω, t) is a stochastic process whose sample functions are, with probability one, right-continuous step-functions in t on the interval [a, b). Then for every ω outside a null set N and every t ɛ [a, b), there is a δ(ω, t) such that < 5(w, t) implies X(ω, t + h) = X(ω, t) which in turn implies (X(ω, t + h) — X(ω, t))/h → 0 with probability one as Since convergence with probability one implies convergence in probability, it follows that the right-hand derivative of X(ω, t) in probability is zero at every t ɛ [a, b). Consequently, if the left-hand t ɛ [a, b), then it is also zero, but in general left-hand derivatives in probability on [a, b) may not even exist as finite-valued random functions. It is also worthwhile to observe that if a process is not differentiable in probability, then it is not differentiable in quadratic mean. From these observations it follows that the claims in Section 6 of the previous paper regarding the existence of non-zero derivatives in quadratic mean are in error, for if a derivative in quadratic mean exists, then it is a derivative in probability and is therefore zero. Incidentally, it turns out that the Poisson process has a zero derivative in probability but is not differentiable in quadratic mean. The fact that the Poisson process has a derivative in probability seems to depend rather crucially on the fact that it is also a process with independent increments.

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

In a previous paper (Moran (1956)) the theory of a dam with a continuous release was developed for a situation in which the input was defined by an additive homogeneous random process of gamma distribution type, and the release was defined to occur continuously at a fixed rate when the content of the dam was non-zero. In the present paper we modify these conditions by assuming first that the input is defined by a general additive homogeneous process with non-negative increments with finite second moment, and secondly that the rate of release is proportional to the content of the dam. This modification to the release rule ensures that the content of the dam is never zero and the theory is then so simplified that the distributional properties of the contents of a sequence of dams in series can be easily found.

This paper arises from the observation by Dirac [1] that an exact measurement of the velocity of an electron should give one of the instantaneous values ± c. The velocity v < c with which the electron moves bodily is in some statistical sense an average. This aspect of the quantum states of motion of a relativistic particle has frequently been mentioned [2], [3] and, as is wellknown, is connected with the self energy problem. Such deductions from quantum mechanics have led the author to consider the possibility of constructing from purely statistical ideas an entity having the physical properties of a relativistic elementary particle. One might reasonably hope to build up characteristics such as rest mass and spin in such a scheme. Properties such as energy and momentum will arise as averages over fluctuating random quantities. In this paper we shall concentrate on rest mass, energy and momentum. We shall show that a zero rest mass object which is regarded as being imbedded in a statistical structure emulates the characteristic of rest mass. The average energy and momentum of this statistical system are then shown to satisfy the usual classical relativistic relation between energy and momentum. The statistical structure chosen is that of an object with a randomly orientated velocity vector, c, which has a fixed magnitude c … |c| where c is the velocity of light.