A learning matrix is defined by a set of input and output pattern vectors. The entries in these vectors are zeros and ones. The matrix is the maximum of the outer products of the input and output pattern vectors. The entries in the matrix are also zeros and ones. The product of this matrix with a selected input pattern vector defines an activity vector. It is shown that when the patterns are taken to be random, then there are central limit and large deviation theorems for the activity vector. They give conditions for when the activity vector may be used to reconstruct the output pattern vector corresponding to the selected input pattern vector.

When does a distribution F have the property of both being in the domain of attraction of exp {–e–x} and having a second convolution-power tail equivalent to the first: Sufficient conditions and examples are given.

We study the sample path properties of χ2 random surfaces, in particular in the neighbourhood of their extrema. We show that, as is the case for their Gaussian counterparts, χ2 surfaces at high levels follow the form of certain deterministic paraboloids, but that, unlike their Gaussian counterparts, at low levels their form is much more random. This has a number of interesting implications in the modelling of rough surfaces and the study of the ‘robustness' of Gaussian field models. The general approach of the paper is the study of extrema via the ‘Slepian model process', which, for χ2 fields, is tractable only at asymptotically high or low levels.

Let {N(t)} be a renewal process having the associated age process {X(t)}. Of interest is the counting process {M(t)} characterized by a non-homogeneous Poisson process with age-dependent intensity function λ (X(t)). The trivariate process {Y(t) = [M(t), N(t), X(t)]} is analyzed obtaining its Laplace transform generating function explicitly. Based on this result, asymptotic behavior of {S(t) = cM(t) + dN(t)} as t → ∞ is discussed. Furthermore, a sufficient condition is given under which {M(t), –N(t), X(t)} is stochastically monotone and associated. This condition also assures increasing stochastic convexity of {M(t)}. The usefulness of these results is demonstrated through an application to the age-dependent minimal repair problem.

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.

This paper introduces the use of symbolic computation (also known as computer algebra) in stochastic analysis and particularly in the Itô calculus. Two related examples are considered: the Clifford-Green theorem on random Gaussian triangles, and a generalization of the D. G. Kendall theorem on the kinematics of shape.

The Clifford–Green theorem gives a remarkable characterization of the joint distribution of the squared-side-lengths of n independent Gaussian points in n-space, namely that this distribution is that of n independent exponential random variables conditioned to satisfy all the inequalities requisite if they are to arise as squared-side-lengths from a point-set in n-space. The D. G. Kendall theorem on the diffusion of shape identifies the statistics of the diffusion arising (under a time-change) as the shape of a triangle whose vertices diffuse by Brownian motion in 2-space or 3-space.

Symbolic Itô calculus is used to give a new proof of the Clifford-Green theorem, and to generalize the D. G. Kendall theorem to the case of triangles in higher-dimensional space whose vertices diffuse either according to Brownian motion or according to an Ornstein–Uhlenbeck process.

We present autoregressive (AR) and autoregressive moving average (ARMA) processes with bivariate exponential (BE) and bivariate geometric (BG) distributions. The theory of positive dependence is used to show that in various cases, the BEAR, BGAR, BEARMA, and BGARMA models consist of associated random variables. We discuss special cases of the BEAR and BGAR processes in which the bivariate processes are stationary and have well-known bivariate exponential and geometric distributions. Finally, we fit a BEAR model to a real data set.

A family of models for discrete-time processes with Poisson marginal distributions is developed and investigated. They have the same correlation structure as the linear ARMA processes. The joint distribution of n consecutive observations in such a process is derived and its properties discussed. In particular, time-reversibility and asymptotic behaviour are considered in detail. A vector autoregressive process is constructed and the behaviour of its components, which are Poisson ARMA processes, is considered. In particular, the two-dimensional case is discussed in detail.

Whittle enunciated an important reduction principle in dynamic programming when he showed that under certain conditions optimal strategies for Markov decision processes (MDPs) placed in parallel to one another take actions in a way which is consistent with the optimal strategies for the individual MDPs. However, the necessary and sufficient conditions given by Whittle are by no means always satisfied. We explore the status of this computationally attractive reduction principle when these conditions fail.

We consider a non-preemptive priority head of the line queueing system with multiple servers and two classes of customers. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. A Markovian state description consists of the number of customers of each class in service and in the queue. We solve a matrix equation to obtain the generating function of the equilibrium probability distribution by analyzing singularities of the equation coefficients, which are meromorphic matrices of two complex variables. We then obtain the mean waiting times for each class.

This paper deals with the single-server queue with batch arrivals. We show that under suitable conditions the waiting-time distribution of an individual customer has an asymptotically exponential expansion. Computationally useful characterizations of the amplitude factor and the decay parameter are given for the practically important case in which the interarrival time and the service time have phase-type distributions.

We consider a queue which is obtained by slightly modifying the M/PH/1 queue, so that weak correlations are introduced among the interarrival times. We show how the stationary probability distribution may be studied by a perturbation analysis.

It is shown by a simple sample path argument that the ruin probabilities for a risk reserve process with premium rate p(r) depending on the reserve r and finite or infinite horizon are related in a simple way to the state probabilities of a compound Poisson dam with the same release rate p(r) at content r. In the infinite horizon case, this result has been established by Harrison and Resnick (1978), and in the finite horizon case with constant p it extends well-known relations to the M/G/1 virtual waiting time.