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…

We prove that an optional process of non-exploding realized power variation along stopping times possesses almost surely làglàd paths. This result is useful for the analysis of some imperfect market models in mathematical finance. In the finance applications variation naturally…

In a one-parameter model for evolution of random trees, which also includes the Barabasi-Albert random tree [1], almost sure behavior and the limiting distribution of the degree of a vertex in a fixed position are examined. A functional central limit…

We study the density of the supremum of a strictly stable Levy process. We prove that for almost all values of the index $\alpha$ - except for a dense set of Lebesgue measure zero - the asymptotic series which were…

We introduce and study a new interacting particles model with a wall and two kinds of interactions - blocking and pushing - which maintain particles in a certain order. We show that it involves a random matrix model.