By making the time parameters of a birth and death process random variables, we create a continuous-time analogue of random walk in a random environment. Criteria for recurrence or transience are discussed and an a.s. convergence law is determined.

It is proved that the r-dimensional volume Vn of the r-simplex spanned by r + 1 sample points taken from the n-fold product distribution of a distribution F is asymptotically normal as n →∞.

We consider a multigrade population with semi-Markov transitions between grades, Poisson arrivals to each grade, and departures from each grade. For this model the joint distribution of the numbers in each grade at any time is found, and the limiting distributions shown to be independently Poisson; this extends a previous result for a multigrade population with Markov transitions and Poisson recruitment.

This model is particularly applicable to manpower planning. The inclusion of semi-Markov transitions allows us to take into account existing knowledge of the distribution of length of service until an individual leaves his firm.

We derive an integral expression for the expected coverage of an n-dimensional target when the value density function of the target, the damage function, and the distribution of the damage function center around the target center are spherically symmetric. This integral expression is transformed by means of Parseval's theorem for Hankel transforms to an integral of the product of the Hankel transforms of order ½n − 1 of all underlying functions.

Given n events A1, A2, · ··, An, bounds are obtained for the probability of the occurrence of at least m out of the n events. The bounds are linear in terms of S1 and S2 where S1 = Σi P(Ai) and S2 = Σi<jΡ (Α i,Α j). The bounds are illustrated with an example. For m = 1 the bounds are the same as the best known linear bounds in terms of S1 and S2.

The forward equations for the death process in ℕ2 with Q-matrix determined by q(i, j; i − 1, j) = αij, i ≧ 1, q(i, j; i, j − 1) = μj, j ≧ 1 are solved by giving a special interpretation to this Q-matrix.

It is pointed out that the conditions given by Haigh (1975) for finding evolutionarily stable strategies corresponding to a given matrix are sufficient, but not always necessary. An example is given, and an amended version of the necessary and sufficient conditions is stated.

A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.

We derive the representation of additive measures of information with the sum property under a regular generating function. This family of measures includes the Shanon entropy, directed/symmetric divergence and inaccuracy.

If the step-length distribution function F for a random walk {Sn, n ≧ 0} is either continuous and symmetric or belongs to the domain of attraction of a symmetric stable law, then it is clear that the symmetric form of ‘Spitzer's condition' holds, i.e. The question considered in this note is whether or not (⋆) can hold for other random walks. The answer is in the affirmative, for we show that (⋆) holds for a large class of random walks for which F is neither symmetric nor belongs to any domain of attraction; all such random walks are asymptotically symmetric, in the sense that limx→∞ {F(–x)| 1 – F(x)} = 1, but we show by an example that this is not a sufficient condition for (⋆) to hold.

Two people independently and with the same distribution guess the location of an unseen object in n-dimensional space, and the one whose guess is closer to the unseen object is declared the winner. The first person announces his guess, but the second modifies his unspoken idea by moving his guess in the direction of the first guess and as close to it as possible. It is shown that if the distribution of guesses is rotationally symmetric about the true location of the unseen object, ¾ is the sharp lower bound for the success probability of the second guesser. If the distribution is fixed and the dimension increases, then for a certain class of distributions, the success probability approaches 1.

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, ….

It is proved that for a Poisson process there exists a one-to-one relation between the distribution of the random variable N(Y) and the distribution of the non-negative random variable Y. This relation is used to characterize the gamma distribution by the negative binomial distribution. Furthermore it is applied to obtain some characterizations of the exponential distribution.

It is shown how to estimate centroids and other measures of location of an unknown embedded subset of a bounded specimen by random sectioning.

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.

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. If the Xi's have a geometric distribution then the conditional distribution of Xk+1:n – Xk:n given Xk+1:n – Xk:n > 0 is the same as the distribution of X1:n–k. Also the random variable X2:n – X1:n is independent of the event [X1:n = 1]. Under mild conditions each of these two properties characterizes the geometric distribution.

A new and simpler proof of Morrison's result that within exponential mixtures only IFR gamma mixing produces linearly increasing mean residual life functions is given. A parallel and new characterization of the DFR gamma laws follows as a consequence. The method of proof used suggests a general result on the infinite divisibility of the mixing distributions in exponential mixtures.

A number of points are chosen at random along a line segment and used as the left end-points of lengths of fixed size. Finite sum expressions are derived for the moments of the portion of the line segment covered by one or more of the lengths. The derivation utilizes a relationship between the coverage and the busy time in an M/D/∞ queue.

On a circle of unit circumference arcs of length a are placed at random. Let Nα be equal to the necessary number of arcs to cover at least the length 1 − p, 0 ≦ p < 1, of the circumference at least m (≧1) times. In the present paper limit distributions of Nα are derived when α → 0. Some results for spacings are also obtained.