We consider the classical Foster-Lyapunov condition for the existence of an invariant measure for a Markov chain when there are no continuity or irreducibility assumptions. Provided a weak uniform countable additivity condition is satisfied, we show that there are a finite number of orthogonal...

A particle system on the real line with singular interaction consisting of electrostatic repulsion and a linear restoring force is considered. The empirical measure process is known to converge weakly in a space of continuous-measure-valued functions. In this work we show that the fluctuations...

Single nucleotide polymorphisms (SNPs) are useful markers for locating genes since they occur throughout the human genome and thousands can be scored at once using DNA microarrays. Here, we use branching processes and coalescent theory to show that if one uses Kruglyak's (Nature Gen. 12 (1999)...

We consider triangular arrays of Markov random walks that can be approximated by an accompanying sequence of diffusion processes. We give uniform bounds for approximation of scaled transition probabilities by transition densities of the diffusion process. In particular, we state local limit...

We employ the interlacing construction to show that the solutions of stochastic differential equations on manifolds which are written in Marcus canonical form and driven by infinite-dimensional semimartingales with jumps give rise to stochastic flows of diffeomorphisms.

Periodogram ordinates of a Gaussian white-noise computed at Fourier frequencies are well known to form an i.i.d. sequence. This is no longer true in the non-Gaussian case. In this paper, we develop a full theory for weighted sums of non-linear functionals of the periodogram of an i.i.d....

We consider the random coloring of the vertices of a graph G, that arises by first performing i.i.d. bond percolation with parameter p on G, and then assigning a random color, chosen according to some prescribed probability distribution on the finite set {0,...,r-1}, to each of the connected...

The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561-574; Schatte, Math. Nachr. 137 (1988) 249-256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular,...

The strategy of bold play in the game of red and black leads to a number of interesting mathematical properties: the player's fortune follows a deterministic map, before the transition that ends the game; the bold strategy can be "re-scaled" to produce new strategies with the same win...

This article presents new results on the problem of selecting (online) a monotone subsequence of maximum expected length from a sequence of i.i.d. random variables. We study the case where the variables are observed sequentially at the occurrence times of a Poisson process with known rate. Our...