Self-similar co-ascent processes and Palm calculus

TL;DR

This paper explores self-similar co-ascent processes, showing they can be characterized via Palm distributions of their record measures, extending the concept beyond stationary processes.

Contribution

It introduces a novel framework linking self-similar co-ascent processes with Palm calculus based on record measures, expanding the theoretical understanding of these processes.

Findings

Co-ascent processes are characterized by Palm distributions of record measures.

The approach extends Palm calculus to self-similar, non-stationary processes.

Provides a new perspective on the structure of self-similar processes.

Abstract

We discuss certain renormalised first passage bridges of self-similar processes. These processes generalise the Brownian co-ascent, a term recently introduced by H. Panzo (S\'eminaire de Probabilit\'es L, 2019). Our main result states that the co-ascent of a given process is the process under the Palm distribution of its record measure. We base our notion of Palm distribution on self-similarity, thereby complementing the more common approach of considering Palm distributions related to stationarity or stationarity of increments of the underlying processes.