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.

Let $B(t)$ be a Brownian motion in $R^3$. A subpath of the Brownian path $B[0,1]$ is a continuous curve $\gamma(t)$, where $\gamma[0,1] \subseteq B[0,1]$ , $\gamma(0) = B(0)$, and $\gamma(1) = B(1)$. It is well-known that any subset $S$ of…