Consider a deck of n cards. Let $p_1,p_2,\ldots,p_n$ be a probability vector and consider the mixing time of the card shuffle which at each step of time picks a position according to the pi 's and move the card in…

We consider an urn model, whose replacement matrix has all entries nonnegative and is balanced, that is, has constant row sums. We obtain the rates of the counts of balls corresponding to each color for the strong laws to hold.…

Branching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where \lq in each generation' the reproduction law is picked randomly in an i.i.d. manner. The associated random walk of the environment has increments distributed like the…

We investigate the asymptotic spectrum of spiked perturbations of Wigner matrices. The entries of the Wigner matrix have a distribution which is symmetric and satisfies a Poincaré inequality. The spectral measure of the deterministic Hermitian perturbation matrix converges to some…

We study an infinite system of independent symmetric random walks on a hierarchical group, in particular, the c-random walks. Such walks are used, e.g., in mathematical physics and population biology. The number variance problem consists in investigating if the variance…

A well known conjecture of Wigner, Dyson, and Mehta asserts that the (appropriately normalized) $k$-point correlation functions of the eigenvalues of random $n \times n$ Wigner matrices in the bulk of the spectrum converge (in various senses) to the $k$-point…

It was shown in Groeneboom (1983) that the least concave majorant of one-sided Brownian motion without drift can be characterized by a jump process with independent increments, which is the inverse of the process of slopes of the least concave…

The paper focuses on spatial multitype branching systems with spatial components (colonies) indexed by a countable group, for example $Z^d$ or the hierarchical group. As type space we allow continua and describe populations in one colony as measures on the…

We consider spatially interacting Moran models and their diffusion limit which are interacting Fisher-Wright diffusions. The Moran model is a spatial population model with individuals of different type located on sites given by elements of an Abelian group. The dynamics…