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We consider random walks on the square lattice of the plane along the lines of Heyde (1982, 1993) and den Hollander (1994), whose studies have in part been inspired by the so-called transport phenomena of statistical physics. Two-dimensional anisotropic random walks with anisotropic density conditions a' la Heyde (1982, 1993) yield fixed column configurations and nearest-neighbour random walks in a random environment on the square lattice of the plane as in den Hollander (1994) result in random column configurations. In both cases we conclude simultaneous weak Donsker and strong Strassen type invariance principles in terms of appropriately constructed anisotropic Brownian motions on the plane, with self-contained proofs in both cases. The style of presentation throughout will be that of a semi-expository survey of related results in a historical context., Comment: 25 pages

We prove that the number gamma(N) of the zeros of a two-parameter simple random walk in its first N-by-N time steps is almost surely equal to N to the power 1+o(1) as N goes to infinity. This is in contrast with our earlier joint effort with Z. Shi [4]; that work shows that the number of zero crossings in the first N-by-N time steps is N to the power (3/2)+o(1) as N goes to infinity. We prove also that the number of zeros on the diagonal in the first N time steps is (c+o(1)) log N as N goes to infinity, where c is 2\pi., Comment: 14 pages

We prove strong invariance principle between a transient Bessel process and a certain nearest neighbor (NN) random walk that is constructed from the former by using stopping times. It is also shown that their local times are close enough to share the same strong limit theorems. It is shown furthermore, that if the difference between the distributions of two NN random walks are small, then the walks themselves can be constructed so that they are close enough. Finally, some consequences concerning strong limit theorems are discussed.

We study some properties of the local time of the asymmetric Bernoulli walk on the line. These properties are very similar to the corresponding ones of the simple symmetric random walks in higher ($d\geq3$) dimension, which we established in the recent years. The goal of this paper is to highlight these similarities.

We prove strong theorems for the local time at infinity of a nearest neighbor transient random walk. First, laws of the iterated logarithm are given for the large values of the local time. Then we investigate the length of intervals over which the walk runs through (always from left to right) without ever returning., Comment: 25 pages

Consider a symmetric aperiodic random walk in $Z^d$, $d\geq 3$. There are points (called heavy points) where the number of visits by the random walk is close to its maximum. We investigate the local times around these heavy points and show that they converge to a deterministic limit as the number of steps tends to infinity., Comment: 19 pages

Given an R^d-valued supercritical branching Wiener process, let D(A,T) be the number of particles in a subset A of R^d at time T, (T=0,1,2,...). We provide a complete asymptotic expansion of D(A,T) as T goes to infinity, generalizing the work of X.Chen.