A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let f Θ,D :I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = f Θ,D (X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

Invariance of a random discrete distribution under size-biased permutation is equivalent to a conjunction of symmetry conditions on its finite-dimensional distributions. This is applied to characterize residual allocation models with independent factors that are invariant under size-biased permutation. Apart from some exceptional cases and minor modifications, such models form a two-parameter family of generalized Dirichlet distributions.

Is the intersection between an arbitrary but fixed plane and the spatial Poisson Voronoi tessellation a planar Voronoi tessellation? In this paper a negative answer is given to this long-standing question in stochastic geometry. The answer remains negative for the intersection between a t-dimensional linear affine space and the d-dimensional Poisson Voronoi tesssellation, where 2 ≦ t ≦ d − 1. Moreover, it is shown that each cell on this intersection is almost surely a non-Voronoi cell.

We solve the Fokker-Planck equation for the Wiener process with drift in the presence of elastic boundaries and a fixed start point. An explicit expression is obtained for the first passage density. The cases with pure absorbing and/or reflecting barriers arise for a special choice of a parameter constellation. These special cases are compared with results in Darling and Siegert [5] and Sweet and Hardin [15].

We study the dispersion of a collection of particles carried by an isotropic Brownian flow in Of particular interest are the center of mass and the centered spatial second moments. Their asymptotic behavior depends strongly on the spatial dimension and the largest Lyapunov exponent of the flow. We use estimates for the pair separation process to give a fairly complete picture of this behavior as t → ∞. In particular, for incompressible flows in two dimensions, we show that the variance of the center of mass grows sublinearly, while dispersion relative to the center of mass grows linearly.

Let X 1, X 2, ·· ·be stationary normal random variables with ρn = cov(X 0, Xn ). The asymptotic joint distribution of and is derived under the condition ρn log n → γ [0,∞). It is seen that the two statistics are asymptotically independent only if γ = 0.

Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

We consider a variation of the M/G/1 queue in which, when the system contains more than k customers, it switches from its initial general service distribution to a different general service distribution until the server is cleared, whereupon it switches back to the original service distribution. Using a technique due to Baccelli and Makowski we define a martingale with respect to an embedded process and from this arrive at a relationship between the process and a modified Markov renewal process. Using this an analysis of the stationary behaviour of the queue is possible.

The object studied in this paper is a pair (Φ, Y), where Φ is a random surface in and Y a random vector field on . The pair is jointly stationary, i.e. its distribution is invariant under translations. The vector field Y is smooth outside Φ but may have discontinuities on Φ. Gauss' divergence theorem is applied to derive a flow conservation law for Y. For this specializes to a well-known rate conservation law for point processes. As an application, relationships for the linear contact distribution of Φ are derived.

The series expansion for the solution of the integral equation for the first-passage-time probability density function, obtained by resorting to the fixed point theorem, is used to achieve approximate evaluations for which error bounds are indicated. A different use of the fixed point theorem is then made to determine lower and upper bounds for asymptotic approximations, and to examine their range of validity.

A Markov chain is used as a model for a sequence of random experiments. The waiting time for sequence patterns is considered. Recursive-type relations for the distribution of waiting times are obtained.

Let N be a stationary Markov-modulated marked point process on ℝ with intensity β ∗ and consider a real-valued functional ψ(N). In this paper we study expansions of the form Eψ(N) = a 0 + β ∗ a 1 + ·· ·+ (β∗ )nan + o((β ∗)n ) for β ∗→ 0. Formulas for the coefficients ai are derived in terms of factorial moment measures of N. We compute a 1 and a 2 for the probability of ruin φ u with initial capital u for the risk process in the Markov-modulated environment; a 0 = 0. Moreover, we give a sufficient condition for ϕu to be an analytic function of β ∗. We allow the premium rate function p(x) to depend on the actual risk reserve.

A homogeneous Gaussian Markov lattice-process model has a regression coefficient that determines the extent to which a random variable of a vertex is dependent on those of the neighbors. In many studies, the absolute value of this parameter has been assumed to be less than the reciprocal of the number of neighbors. This condition is shown to be necessary and sufficient for the existence of the Gaussian process satisfying the model equations under some assumptions on lattices using the notion of dual processes. We also give examples of models that neither satisfy the condition imposed on the region for the parameter nor the assumptions on lattices. A formula for autocovariance functions of Gaussian Markov processes on general lattices is derived, and numerical procedures to calculate the autocovariance functions are proposed.

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li ) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li . In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

We consider the exact distribution of the number of peaks in a random permutation of the integers 1, 2, ···, n. This arises from a test of whether n successive observations from a continuous distribution are i.i.d. The Eulerian numbers, which figure in the p.g.f., are then shown to provide a link between the simpler problem of ascents (which has been thoroughly analysed) and both our problem of peaks and similar problems on the circle. This link then permits easy deduction of certain general properties, such as linearity in n of the cumulants, in the more complex settings. Since the focus of the paper is on exact distributional results, a uniform bound on the deviation from the limiting normal is included. A secondary purpose of the paper is synthesis, beginning with the more familiar setting of peaks and troughs.

Grenander et al. (1991) proposed a conditional cyclic Gaussian Markov random field model for the edges of a closed outline in the plane. In this paper the model is recast as an improper cyclic Gaussian Markov random field for the vertices. The limiting behaviour of this model when the vertices become closely spaced is also described and in particular its relationship with the theory of ‘snakes' (Kass et al. 1987) is established. Applications are given in Grenander et al. (1991), Mardia et al. (1991) and Kent et al. (1992).

In this paper we continue the study of the classification problem for random walks in the quarter plane, with zero drifts in the interior of the domain. The necessary and sufficient conditions for these random walks to be ergodic were found earlier in [3]. Here we obtain necessary and sufficient conditions for transience by constructing suitable Lyapounov functions.

In this paper we give a solution of an optimal stopping problem concerning random walks with non-zero drift, thereby proving the necessity of the existence of ESτ for Wald's equation ES τ = ES 1 · Ετ to hold, even if attention is restricted to non-randomized stopping times τ. This answers a question of Robbins and Samuel (1966).

We study the limiting behaviour of large systems of two types of Brownian particles undergoing bisexual branching. Particles of each type generate individuals of both types, and the respective branching law is asymptotically critical for the two-dimensional system, while being subcritical for each individual population.

The main result of the paper is that the limiting behaviour of suitably scaled sums and differences of the two populations is given by a pair of measure and distribution valued processes which, together, determine the limit behaviours of the individual populations.

Our proofs are based on the martingale problem approach to general state space processes. The fact that our limit involves both measure and distribution valued processes requires the development of some new methodologies of independent interest.