In the study of bivariate processes, one often encounters expressions involving repeated combinations of bivariate continuum operations such as multiple bivariate convolutions, marginal convolutions, tail integration, partial differentiation and multiplication by bivariate polynomials. In many cases numerical computation of such results is quite tedious and laborious. In this paper, the bivariate Laguerre transform is developed which provides a systematic numerical tool for evaluating such bivariate continuum operations. The formalism is an extension of the univariate Laguerre transform developed by Keilson and Nunn (1979), Keilson et al. (1981) and Keilson and Sumita (1981), using the product orthonormal basis generated from Laguerre functions. The power of the procedure is proven through numerical exploration of bivariate processes arising from correlated cumulative shock models.

Using a representation formula expressing the mixed cumulants of realvalued random variables by corresponding moments, sufficient conditions are given for the normal convergence of suitably standardized shot noise assuming that the generating stationary point process is independently marked and Brillinger mixing and that its intensity tends to ∞. Furthermore, estimates for the rate of this normal convergence are obtained by exploiting a general lemma on probabilities of large deviations and on the rate of normal convergence.

For each N≧1, let {XN(t, x), t≧0} be a discrete-time stochastic process with XN (0) = x. Let FN (y) = E(XN (t + 1) | XN (t) = y), and define YN (t, x) = FN(YN(t – 1, x)), t≧1 and YN (0, x) = x. Assume that in a neighborhood of the origin FN (y) = mNy(l + O(y)) where mN > 1, and define for δ> 0 and x> 0, υN (δ, x) = inf{t:xmtN >δ}. Conditions are given under which, for θ> 0 and ε> 0, there exist constants δ > 0 and L <∞, depending on εand 0, such that This result together with a result of Kurtz (1970), (1971) shows that, under appropriate conditions, the time needed for the stochastic process {XN (t, 1/N), t≧0} to escape a δ -neighborhood of the origin is of order log Νδ /log mN . To illustrate the results the Wright-Fisher model with selection is considered.

Let T be a triangle. P be a parallelogram, E be an ellipse, A , B be concentric circles, C , D be concentric dartboard regions, R , S be rectangles of the same orientation, U, V be two finite unions and/or differences of convex regions in the Euclidean plane. Given a function f on [0,∞), let E[/(r ), U , V ] denote the mean value of f(| u –v |), where |u –v | is the distance between u ∊U and v ∊V . Using Borel’s overlap technique, a specific distance weight function and a specific equivalence relation, we obtain formulae expressing E[f(r ), U , V ] in terms of triple integrals, expressing E(r n , U , V ), E[f(r ), A , V ] and E[f(r ), R , V ] in terms of double integrals, expressing E[f(r ), A , B ], E[f(r ), R , S ], E[f(r ), T , T ], E[f(r ), P , P ], E(rn , C , D ) and E(r n , R , V ) in terms of single integrals, and expressing E(r n , R , S ), E(r n , P , P ), E(r n , T , T ), E(r n , E , E ) in terms of elementary functions, where n is an integer ≧−1. Many other related results are also given.

The Radon transform of a plane domain is a random variable assigning to each line in the plane the chord length of its intersection with the domain. The probability distribution of this random variable does not characterize the domain, but it is shown to characterize a sufficiently asymmetric convex polygon. Under weaker assumptions, a convex polygon is characterized by this distribution, up to a finite number of rearrangements.

Consider a Poisson point process of density 1 in R d, centered so that the origin is one of the points. Using lv distances, 1≦p≦∞, define Nd as the number of other points which have the origin as their nearest neighbor and Vol Vd as the volume of the Voronoi region of the origin. We prove that Nd → Poisson (λ = 1) and Vol Vd → 1 in distribution as d →∞, thus extending previous results from the case p = 2. More generally, for a variety of exchangeable distributions for n + 1 points, e 0, · ··, e n, in Rd and a variety of distances, we obtain the asymptotic behavior of Ndn , the number of points which have e 0 as their nearest neighbor, as n, d → ∞ in one or both of the possible iterated orders. The distributions treated include points distributed on the unit l2 sphere and the distances treated include non-l p distances related to correlation coefficients.

The maximum-entropy approach to the estimation of the spectral density of a time series has become quite popular during the last decade. It is closely related to the fact that an autoregressive process of order p has maximal entropy among all time series sharing the same autocovariances up to lag p. We give a natural generalization of this result by proving that a mixed autoregressive-moving-average process (ARMA process) of order (p, q) has maximal entropy among all time series sharing the same autocovariances up to lag p and the same impulse response coefficients up to lag q. The latter may be estimated from a finite record of the time series, for example by using a method proposed by Bhansali (1976). By the way, we give a result on the existence of ARMA processes with prescribed autocovariances up to lag p and impulse response coefficients up to lag q.

The aim of this paper is to present a few techniques which may be useful in the analysis of time series when a failure is suspected. We present two categories of tests and investigate their asymptotic properties: one, of nonparametric type, is intended to detect a general failure in spectrum; the other investigates the properties of likelihood ratio tests in parametric models which have a non-standard behaviour in this situation. Finally, we obtain the asymptotic distribution of the likelihood estimators of the change parameters.

The concept of a characterization and its stability for queueing models is introduced. The principle of two stages in the study of the stability property is formulated. A series of results concerning the G/G/1 model is obtained.

A relation between the distributions of the workload of an arbitrary server and the waiting time is obtained for a GI/G/s queue in steady state. An expression for the total workload of the system (total unprocessed work) is also found. Several well-known results emerge as special cases. Various useful relations, for instance bounds and approximations for the mean waiting time, are also given.

It is proved that a distribution F is strongly unimodal iff any two quantiles of the convolution of F with any other distribution are further apart than the corresponding quantiles of F itself. Some related characterizations of strong unimodality are also given.

Asymptotic series are derived for two integrals using a Gaussian identity and Laplace’s method, demonstrating an improvement over earlier methods.

The aim of this note is to prove the following characterization.

Bounds obtained by Ramalhoto [1] for the variance of the busy period of an M/G/∞ queue are improved.