Almost periodicity as a path property for $p$-adic self-similar processes with stationary increments

TL;DR

This paper investigates various notions of almost periodicity in $p$-adic self-similar processes with stationary increments, establishing key equivalences and extensions to random fields.

Contribution

It characterizes Bohr almost periodicity as path continuity in $p$-adic processes and extends results to finite-dimensional random fields.

Findings

Bohr almost periodicity is equivalent to $p$-adic path continuity.

Weyl and Besicovitch almost periodicity do not share this equivalence.

Results are extended to finite-dimensional random fields.

Abstract

Shen and Zhang (2021) showed that almost periodicity naturally arises in the spectral representation of discrete-time $p$-adic self-similar processes with stationary increments. In this paper, we study several notions of almost periodicity as sample path properties of Banach space-valued $p$-adic sssi processes. We prove that Bohr almost periodicity is equivalent, as a path event, to continuity with respect to the $p$-adic topology. We also show that the corresponding equivalence fails for Weyl and Besicovitch almost periodicity. Finally, we extend the Bohr almost-periodic result to finite-dimensional random fields.