Abstract: We study properties of the trajectories of a random walk with independent increments whose jump distribution alternates between F1 and F2 according to the following rule. The distribution F1, which has a positive mean, is retained until the trajectory first reaches the set (−∞,−a), a ≥ 0, after which the process switches to a random walk with negative drift corresponding to the distribution F2. This continues until the trajectory first reaches the horizontal boundary at level b ≥ 0, after which the jump distribution switches back to F1, and so on. We give estimates for the probabilities that the switching times are finite and for the probability that the supremum of the trajectory is infinite, as well as exact formulas for these characteristics in some special cases. In the general case, we obtain double transforms of the joint distributions of the switching times and the positions of the process at these times, as well as asymptotic representations for them as a →∞, b →∞under Cram´er-type conditions on the distributions F1 and F2.

Cite: Lotov V.I.
On a Random Walk with a Specific Switching Algorithm
Siberian Mathematical Journal. 2026. V.67. N5. P.1164-1179. DOI: 10.1134/s0037446626050162 WOS Scopus OpenAlex

Original: Лотов В.И.
О случайном блуждании со специфическим алгоритмом переключений
Сибирский математический журнал. 2026. Т.67. №5. С.960-978. DOI: 10.33048/smzh.2026.67.516

Dates:

Submitted:	Apr 8, 2026
Accepted:	Apr 13, 2026
Published print:	Sep 24, 2026
Published online:	Sep 24, 2026

Identifiers:

≡ Web of science:	WOS:001885470600015
≡ Scopus:	2-s2.0-105051656242
≡ OpenAlex:	W7214208269

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