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Various notions of geometric ergodicity for Markov chains on general state spaces exist. In this paper, we review certain relations and implications among them. We then apply these results to a collection of chains commonly used in Markov chain Monte…

In this note the almost sure convergence of stationary, $\varphi$-mixing sequences of random variables with values in real, separable Banach spaces according to summability methods is linked to the fulfillment of a certain integrability condition generalizing and extending the results…

The trajectory of the ball in a soccer game is modelled by the Brownian motion on a cylinder, subject to elastic reflections at the boundary points (as proposed in [KPY]). The score is then the number of windings of the…

We prove the law of large numbers for $U$-statistics whose underlying sequence of random variables satisfies an absolute regularity condition ($\beta$-mixing condition) under suboptimal conditions.

We show that if a sequence of domains $D_k$ increases to a domain $D$ then the reflected Brownian motions in $D_k$'s converge to the reflected Brownian motion in $D$, under mild technical assumptions.

In this short note we show that the paths two independent loop-erased random walks in four dimensions intersect infinitely often. We actually prove the stronger result that the cut-points of the two walks intersect infinitely often.

We sharpen a bound in a theorem of Russell Lyons for percolation on a tree and associated random walk.

Noise sensitivity of functions on the leaves of a binary tree is studied, and a hypercontractive inequality is obtained. We deduce that the spider walk is not noise stable.

Recently, Kalvin M. Jansons derived in an elegant way the Laplace transform of the time spent by an excursion above a given level $a>0$. This result can also be derived from previous work of the author on the occupation time…