The problem of estimating an unknown compact convex set K in the plane, from a sample (X1,···,Xn) of points independently and uniformly distributed over K, is considered. Let Kn be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ (K, Kn). Under mild conditions, limit laws for Δn are obtained. We find sequences (an), (bn) such that

(Δn - bn)/an → Λ (n → ∞), where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.

A stochastic dynamical context is developed for Bookstein's shape theory. It is shown how Bookstein's shape space for planar triangles arises naturally when the landmarks are moved around by a special Brownian motion on the general linear group of invertible (2×2) real matrices. Asymptotics for the Brownian transition density are used to suggest an exponential family of distributions, which is analogous to the von Mises-Fisher spherical distribution and which has already been studied by J. K. Jensen. The computer algebra implementation Itovsn3 (W. S. Kendall) of stochastic calculus is used to perform the calculations (some of which actually date back to work by Dyson on eigenvalues of random matrices and by Dynkin on Brownian motion on ellipsoids). An interesting feature of these calculations is that they include the first application (to the author's knowledge) of the Gröbner basis algorithm in a stochastic calculus context.

Let us consider, in the Euclidean space En, a fixed n-dimensional convex body K0 of volume V0 and a system K1,…,Km of mn-dimensional convex bodies, congruent to a convex set K. Assume that the sets Ki (i = 1,…,m) have random positions, being stochastically independent and uniformly distributed on a limited domain of En and denote by Vm the volume of the convex body Km = K0 ∩ (K1 ∩ … ∩ Km). The aim of this paper is the evaluation of the second moment of the random variable Vm.

Consider a network whose nodes are the integer lattice points and whose arcs are fuses of 1Ω resistance. Remove a horizontal segment of N adjacent vertical arcs, forming a ‘crack’ of length N. Subject the network to a uniform potential gradient of v volts per arc in the north-south (or vertical) direction and measure the current in one of the two vertical arcs at the ends of the crack. We write this current in the form e(N)v, and call e(N) the current enhancement.

We show that the enhancement grows at a rate that is the order of the square root of the crack length. Our method is to identify the enhancement with the mean time to exit an interval for a certain integer valued random walk, and then to use some of the well-known Fourier methods for studying random walk. Our random walk has no mean or higher moments and is in the domain of attraction of the Cauchy law. We provide a good approximation to the enhancement using the explicitly known mean time to exit an interval for a Cauchy process. Weak convergence arguments together with an estimate of a recurrence probability enable us to show that the current in an intact fuse, that is in the interior of a crack of length N, grows p roportionally with N/logN.

In this paper, we study Markov chains with infinite state block-structured transition matrices, whose states are partitioned into levels according to the block structure, and various associated measures. Roughly speaking, these measures involve first passage times or expected numbers of visits to certain levels without hitting other levels. They are very important and often play a key role in the study of a Markov chain. Necessary and/or sufficient conditions are obtained for a Markov chain to be positive recurrent, recurrent, or transient in terms of these measures. Results are obtained for general irreducible Markov chains as well as those with transition matrices possessing some block structure. We also discuss the decomposition or the factorization of the characteristic equations of these measures. In the scalar case, we locate the zeros of these characteristic functions and therefore use these zeros to characterize a Markov chain. Examples and various remarks are given to illustrate some of the results.

We consider Brownian motion with a negative drift conditioned to stay positive. We give a sufficient condition for an initial measure to be in the domain of attraction of a quasi-stationary distribution. We construct a counter-example that strongly suggests that this condition is optimal.

Equipping the edges of a finite rooted tree with independent resistances that are inverse Gaussian for interior edges and reciprocal inverse Gaussian for terminal edges makes it possible, for suitable constellations of the parameters, to show that the total resistance is reciprocal inverse Gaussian (Barndorff-Nielsen 1994). This result is extended to infinite trees. Also, a connection to Brownian diffusion is established and, for the case of finite trees, an exact distributional and independence result is derived for the conditional model given the total resistance.

The shift method consists in computing the expectation of an integrable functional F defined on the probability space ((ℝd)N, B(ℝd)⊗N, μ⊗N) (μ is a probability measure on ℝd) using Birkhoff's Pointwise Ergodic Theorem, i.e.

as n → ∞, where θ denotes the canonical shift operator. When F lies in L2(FT, μ⊗N) for some integrable enough stopping time T, several weak (CLT) or strong (Gàl-Koksma Theorem or LIL) converging rates hold. The method successfully competes with Monte Carlo. The aim of this paper is to extend these results to more general probability distributions P on ((ℝd)N, B(ℝd)⊗N), namely when the canonical process (Xn)n∊N is P-stationary, α-mixing and fulfils Ibragimov's assumption

for some δ > 0. One application is the computation of the expectation of functionals of an α-mixing Markov Chain, under its stationary distribution Pν. It may both provide a better accuracy and save the random number generator compared to the usual Monte Carlo or shift methods on independent innovations.