A generating function technique is used to determine the probability that the deviation between two empirical distributions drawn from the same population lies within a given band a specified number of times. We also treat the asymptotic problem of very large sample size, and obtain explicit expressions when the relative number of failures is very small or very large.

Khintchine (1938) showed that a real random variable Z has a unimodal distribution with mode at 0 iff Z ~ U X (that is, Z is distributed like U X), where U is uniform on [0, 1] and U and X are independent. Isii ((1958), page 173) defines a modified Stieltjes transform of a distribution function F for w complex thus: Apparently unaware of Khintchine's work, he proved (pages 179-180) that F is unimodal with mode at 0 iff there is a distribution function Φ for which . The equivalence of Khintchine's and Isii's results is made vivid by a proof (due to L. A. Shepp) in the next section.

L'enveloppe convexe des nuages de points dans Rn a donné lieu à un certain nombre de travaux. Dans deux articles ([5] et [6]), Rényi et Sulanke ont proposé et résolu un certain nombre de problèmes. Une loi de probabilité étant donnée dans R2, on procède aux tirages indépendants de N points suivant cette loi.

In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation used to describe a number of processes including radioactive decay and population growth.

Various formulas of Wald relating to randomly stopped sums have well known continuous-time analogs, holding in particular for Wiener processes. However, sufficiently general forms of most of these do not appear explicitly in the literature. Recent papers by Robbins and Samuel (1966) and by Brown (1969) provide general results on Wald's equations in discrete time and these are here extended (Theorems 2 and 3) to Wiener processes and other homogeneous additive processes, that is, continuous-time processes with stationary independent increments. We also give an inequality (Theorem 1) related to Wald's identity in continuous time, and we derive, as corollaries of Wald's equations, bounds on the variance of an arbitrary stopping time. The Wiener process versions of these results find application in a variety of stopping problems. Specifically, all are used in Hall ((1968), (1969)); see also Bechhofer, Kiefer, and Sobel (1968), Root (1969), and Shepp (1967).

The purpose of this note is two-fold. First, to introduce the mathematical reader to a group of problems in the study of language change which has received little attention from mathematicians and probabilists. Secondly, to introduce a birth and death process, arising naturally out of this group of problems, which has received little attention in the literature. This process can be solved using the standard methods and the solution is exhibited here.

A population which develops as a linear birth process from one individual gives rise to a family tree of size N which shows the relationships of the N members to each other. If this tree is given, but with no a priori information of which point is the root (the original member) we seek to utilize our knowledge of the stochastic manner in which the tree grew to estimate this root. A maximum likelihood estimate is obtained, and the probabilities that this estimate is the true kth point, or a member of the true rth generation are obtained as functions of N. As N → ∞ these probabilities converge to non-zero values for any fixed k, r, which are surprisingly large—for example, when k is 1, the limiting value is 1 — loge2 ≈ .30685.

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

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

Suppose that in a ballot candidate A scores a votes and candidate B scores b votes and all the possible voting records are equally probable. Denote by αr and βr the number of votes registered for A and B respectively among the first r votes counted. For j= 0, 1,···, a + b define Pj(a, b) as the probability that the inequality αr > aβr/b holds for exactly j subscripts r = 1, 2, .., a + b.

Let P= {p1,···, PN} be a finite discrete probability distribution. Then the entropy of the distribution P, introduced by Shannon [12], is defined as Throughout this paper ∑ will stand for and logarithms will be taken to the base D (D > 1).

The fundamental principle underlying insurance is that the expected value of claims is equal to the premium. This was established by Bernoulli [4] in 1738. Subsequent work on the development of ‘risk theory’ led to research concerning the probability of ‘ruin’ of an insurance company. A critical study of these investigations was made by Borch [3] in a recent paper.

Consider an airplane which is at a distance t ≧ 0 from its destination (the distance being measured in time-units) and has n anti-aircraft missiles. The enemy sends its aircraft according to a Poisson process with known parameter, which by redefining the time-scale may, without loss of generality, be taken to be one. When meeting an enemy aircraft, the plane can use, simultaneously, some of its missiles, and it is assumed that the hits are independent random events with known probability 1 > 1 – q > 0. If no missile hits the enemy aircraft, the airplane is shot down with known probability 1 ≧ 1 – u > 0. The case u= 0 is of particular interest. (We have excluded the values q = 0, q = 1 and u= 1 since these yield trivial solutions.)

Some inequalities are obtained which yield bounds for the mean life of series and of parallel systems in the case where component life distributions have properties such as a monotone failure rate, monotone failure rate average, or decreasing density. These bounds are based on comparisons with systems of exponential or uniform components. Similar comparisons are obtained when components have Weibull or Gamma distributions with different shape parameters. Some inequalities are also obtained for convolutions of life distributions helpful in the study of replacement policies.

In the type II counter with constant deadtime, particles which arrive within some constant time τ following another particle are unrecorded. We can think of this process as an alternating sequence of gaps and bunches of events. Gaps have duration > τ, while the intervals between any successive pair of events within a bunch are all ≦ τ. Counter theory is usually concerned with the distribution of intervals between recorded events (i.e., the first event of each bunch) and the distribution of the number of recorded events in a given time interval. In the case where the events form a renewal process this has been studied intensively by Pyke [2], Smith [5] and Takács [6].

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

The queueing system with alternating priorities has been discussed recently in numerous articles (Maxwell [1], Avi Itzhak, Maxwell, Miller [2], Neuts, Yadin [3], and others). Consider a system which consists of several queueing units. Each unit is independent in the sense that it has its own waiting line which is generated by an independent Poisson stream of customers. However, all of the units are served by a single server who allocates his time between the units.

In this paper it is shown that the distribution of the instant of service of a customer is symmetric as between the distributions of service and interarrival time. Also U(t), the expected number of departures in (0, t), is a delayed renewal function for the GI/M/1 queue.

We consider a single server queueing system with inter-arrival times {un, n ≧ 1}, and service times {υn, n ≧ 1} and the queue-discipline, ‘first-come, first-served’. It is assumed that {un} and {υn} are two independent renewal processes, and 0 > E(un) =a < ∞, 0 < E(υn) = b < ∞. The traffic intensity is P ρ b/a(0 > ρ > ∞). This paper is concerned with the case ρ = 1, where it is known that the various queueing processes such as the queue-length Q(t) and waiting time W(t) diverge to + ∞ in distribution as t → ∞. Borovkov [1], [2] and Brody [3] have obtained limit distributions for Q(t) and W(t) with appropriate location and scale parameters in the cases P ≧ 1. Here we investigate random variables related to the busy and idle periods in the system. To explain our approach, we consider the random variables Xn = υn – un (n ≧ 1). Let S0 ≡ 0, Sn = X1 + X2 + ··· + Xn (n ≧ 1), and define the sequence {Nk, k ≧ 0} as

This note describes a model for inflow-stock-outflow time series which are formed as sums of random numbers of subcomponents of randomly determined sizes and are related via a probabilisticly distributed lag. Such time series may be illustrated by population processes in connection with which not only numbers of units are of interest but also the sizes of the units and the sums of the sizes. Such series may also be illustrated by economic phenomena like new loans made, loans outstanding and loans repaid of a commercial bank; new capital investments, capital stock and depreciated capital for a group of firms as well as accessions to the labor force, labor force outstanding, and secessions from the labor force.