Abstract: We study some properties of trajectories of a random walk, with independent summands, for which the distribution of jumps changes according to the following rule. The distribution of jumps with positive expectation is preserved until the trajectory first reaches the lower half-plane, after which a switch to a walk with negative drift occurs. This continues until the trajectory first reaches the upper half-plane, after which it resumes a walk with positive drift, and so on. The paper presents estimates for the probabilities of finiteness of the moments of intersection of the abscissa axis by the random walk trajectory and for the probability of the trajectory’s supremum being infinite. It is found that the infinity of the trajectory’s supremum excludes the simultaneous minus infinity of the trajectory’s infimum. The possibility of finding exact expressions for the characteristics under study is discussed, and examples of their calculation in certain situations are given.

Cite: Lotov V.I.
On a Model of Oscillating Random Walk
Mathematical Notes. 2026. V.119. N5-6. P.930-938. DOI: 10.1134/s0001434626602765 Scopus OpenAlex

Dates:

Submitted:	Aug 1, 2025
Accepted:	Dec 20, 2025
Published print:	Jul 26, 2026

Identifiers:

≡ Scopus:	2-s2.0-105045650951
≡ OpenAlex:	W7171294747

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