Level numbers preserving transformations on excursionsdefined by random walks with state dependent jump laws

TL;DR

This paper investigates the properties of excursions of state-dependent random walks, focusing on level number preservation under transformations, and characterizes the transformations that maintain the probability law of these excursions.

Contribution

It introduces a detailed analysis of level number preserving transformations and characterizes the class of transformations that preserve excursion probabilities.

Findings

Number of excursions with fixed level numbers computed.

Shifts generate all excursions with the same level numbers.

Level numbers behavior analyzed under Vervaat and Doob transforms.

Abstract

We study the number of individuals per level defined by excursions of random walks with state dependent jump law. These level numbersdetermine the probability of the excursion, and the set of transformations preserving the level numbers is, generically, the set of transformation that preserve the probability law of excursions.We compute the number of excursions having a fixed level numbers and we show that the class of shifts of excursions generate all the excursions having a fixed level numbers. We study the behaviorof the level numbers under the Vervaat transform and the Doob transform.