For a suitable definition of the local time of a random walk strong invariance principles are proved, saying that this local time is like that of a Wiener process. Consequences of these results are LIL statements for the local time of a general enough class of random walks. One of the tools for...

We establish moduli of continuity and large increment properties for stationary increment Gaussian processes in order to study the path behavior of infinite series of independent Ornstein-Uhlenbeck processes. The existence and continuity of the latter infinite series type Gaussian processes are...

We give strong results for the maximal number of points of a d-dimensional Poisson process in cubes of volume VT contained in [0, T]d for the case where VT is below the Erdös-Réyni range, i.e. VT = o(log T).

{W(x, y), x>=0, y>=0} be a Wiener process and let [eta](u, (x, y)) be its local time. The continuity of [eta] in (x, y) is investigated, i.e., an upper estimate of the process [eta]([mu], [x, x + [alpha]) - [y, y + [beta])) is given when [alpha][beta] is small.

A particle system on d is considered whose evolution is as follows. At each unit of time each particle independently is replaced by a new generation. The size of a new generation descending from a particle at site x has a distribution and each of its members independently jump to a neighbouring...

The area of the largest circle around the origin completely covered by a simple symmetric plane random walk is investigated.

Let X1, X2... be a sequence of positive, independent, identically distributed (i.i.d.) random variables with S0 = 0, Sn = X1 + ... + Xn, n [greater-or-equal, slanted] 1. Denote by [tau]i = sup{nSn [less-than-or-equals, slant] t }. In this paper we establish almost sure lower and upper bounds for...