The general part of the Borel-Cantelli lemma says that for any sequence of events (An) defined on a probability space (Ω, Σ, P), the divergence of ΣnP(An) is necessary for P(An i.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the An are independent. This paper provides a simple counterpart of this lemma in the sense that the independence condition is replaced by for some . We will see that this property of (An) may frequently be assumed without loss of generality. We also disclose a useful duality which allows straightforward conclusions without selecting independent sequences. A simple random walk example and a new result in the theory of ϕ -branching processes will show the tractability of the method.

It is proved that under certain conditions the probability of explosion of a first point process is not larger than the probability of explosion of a second point process if the intensities of both processes fulfil the same inequality.

It is well known that the number of renewals in the time interval [0, t] for an ordinary renewal process is approximately normally distributed under general conditions. We give a remainder term estimate for this normal distribution approximation.

We offer a short proof that the bridging condition introduced by Whittle is sufficient for regularity in negative dynamic programming. We exploit concavity of the optimal value operator and do not need a special treatment of the case when optimal policies do not exist.

A new markovian process {Xi : i = 0, 1, 2, ·· ·} following a negative exponential distribution and with the same autocorrelation function as the lag-1 autoregressive process is proposed and studied in this paper. The exact distribution of the maxima and of the minima of n consecutive Xi values are obtained and the exact expected upcrossing interval is given for any crossing level.

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.

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.

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.