Between 2007 and 2018, we collaborated extensively with Pál Révész and Miklós Csörgő on many of the problems discussed in this paper. Over the past six years, we have continued to explore these issues, and here, we present some of the most intriguing open questions in these areas. This paper compiles key results from a dozen of our previous works, providing the necessary background to frame these compelling unresolved questions.

We study the path behavior of the symmetric walk on some special comb-type subsets of ℤ 2 which are obtained from ℤ 2 by generalizing the comb having finitely many horizontal lines instead of one.

We study the local time of the anisotropic random walk on the two-dimensional lattice Z(2), by establishing the exact asymptotic behavior of the N-step return probability to the origin.

A tribute to the life and work of Pal Revesz. The Hungarian mathematical community lost one of his leading members, when Pal Revesz passed away on 14 of November 2022.

Generalizing results of Schatte [11] and Atlagh and Weber [2], in this paper we give conditions for a sequence of random variables to satisfy the almost sure central limit theorem along a given sequence of integers.

The Half-Plane Half-Comb walk is a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We prove that the probability that this walk returns to the origin in 2N steps is asymptotically equal to $$2/(\pi N).$$ As a consequence, we prove strong laws and a limit distribution for the local time.

We study the path behavior of the anisotropic random walk on the two-dimensional lattice Z^2. Simultaneous strong approximations of its components are given.

A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs, we establish limit theorems for n-step local and occupation times.

We study the path behavior of the simple symmetric walk on some comb-type subsets of Z2which are obtained from Z2 by removing all horizontal edges belonging to certain sets of values on the y-axis. We obtain some strong approximation results and discuss their consequences.

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.

We consider two or more simple symmetric walks on some graphs, e.g. the real line, the plane or the two dimensional comb lattice, and investigate the properties of the distance among the walkers.

A simple random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition probabilities are studied, and for a fixed number of legs we investigate how high the walker and the Brownian motion can go on the legs in n steps. The heights on the legs are also investigated when the number of legs goes to infinity.

We study the asymptotic behaviour of partial sums of long range dependent random variables and that of their counting process, together with an appropriately normalized integral process of the sum of these two processes, the so-called Vervaat process. The first two of these processes are approximated by an appropriately constructed fractional Brownian motion, while the Vervaat process in turn is approximated by the square of the same fractional Brownian motion.

We study the asymptotic behaviour of Bahadur-Kiefer processes that are generated by summing partial sums of (weakly or strongly dependent) random variables and their renewals. Known results for i.i.d. case will be extended to dependent cases.

This is an expository paper on the asymptotic results concerning path behaviour of the anisotropic random walk on the two-dimensional square lattice Z^2. In recent years Miklós and the authors of the present paper investigated the properties of this random walk concerning strong approximations, local times and range. We give a survey of these results together with some further problems.

We study the path behaviour of the anisotropic random walk on the two-dimensional lattice ℤ2. Strong approximation of its components with independent Wiener processes is proved. We also give some asymptotic results for the local time in the periodic case.

We study limiting properties of a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We give strong approximations for the components with random time changed Wiener processes. As consequences, limiting distributions and some laws of the iterated logarithm are presented. Finally, a formula is given for the probability that the random walk returns to the origin in 2N steps.

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C2 that is obtained from Z2 by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.