We characterize the upper and lower functions of a real-valued Wiener process normalized by the supremum of its local times.

We prove a strong approximation for the spatial Kesten-Spitzer random walk in random scenery by a Wiener process.

We prove that the number Z(N) of level crossings of a two-parameter simple random walk in its first NxN steps is almost surely N3/2+o(1) as N--[infinity]. The main ingredient is a strong approximation of Z(N) by the crossing local time of a Brownian sheet. Our result provides a useful algorithm...

We consider a transient random walk on in random environment, and study the almost sure asymptotics of the supremum of its local time. Our main result states that if the random walk has zero speed, there is a (random) sequence of sites and a (random) sequence of times such that the walk spends a...

We obtain some liminf limits for the Wiener sheet. The approach relies on a careful analysis of the lower tail of the Ornstein-Uhlenbeck process. Our results can be applied to normalized Kiefer and empirical processes. In particular, they yield a satisfying answer to Hirsch's problem for the...

For one-dimensional simple random walk in a general i.i.d. scenery and its limiting process, we construct a coupling with explicit rate of approximation, extending a recent result for Gaussian sceneries due to Khoshnevisan and Lewis. Furthermore, we explicitly identify the constant in the law of...

Our main intention is to describe the behavior of the (cumulative) distribution function of the random variable M0,1 := sup0[less-than-or-equals, slant]s,t[less-than-or-equals, slant]1 W(s,t) near 0, where W denotes one-dimensional, two-parameter Brownian sheet. A remarkable result of Florit and...

We study the small deviation problem for a class of symmetric Lévy processes, namely, subordinated Lévy processes. These processes can be represented as WoA, where W is a standard Brownian motion, and A is a subordinator independent of W. Under some mild general assumption, we give precise...

We present an accurate description for the location of maximum of d-dimensional Brownian motion. In case d = 1, this is a well-known theorem of Csáki et al. (1987a). We also deduce, as application, a version of the iterated logarithm law for the favourite site of transient Brownian motion.

In random environments, the most elementary processes are Sinai's simple random walk and Brox's diffusion process, respectively in discrete and continuous time settings. The two processes are often considered as a kind of companions, somewhat in the same way as the usual random walk and Brownian...