Abstract:In this paper we study the number of returns to the coordinate axes for two-dimensional nearest-neighbour random walks. While one-dimensional results on returns are classical, much less is known in higher dimensions. We analyse the asymptotic behaviour of returns under several natural conditionings: the unconditioned walk, bridges, meanders, and non-negative bridges (or excursions). Our main results characterize the limiting distributions under appropriate rescaling. The resulting one-dimensional marginals may be half-normal, Rayleigh, geometric, negative binomial, or certain mixtures thereof. In most situations, the coordinates are asymptotically independent; however, there are notable exceptions for the meander case, depending on the drift. The proofs rely on conditioning on the numbers of horizontal and vertical steps, which restores a form of independence and reduces the problem to one-dimensional estimates via binomial convolution and Bernstein-type approximations.
From: Kilian Raschel [view email]
[v1]
Mon, 22 Dec 2025 19:24:22 UTC (25 KB)
[v2]
Tue, 1 Sep 2026 15:35:05 UTC (31 KB)
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