Bochner-Almost Periodicity for Stochastic Processes

Abstract

We compare several notions of almost periodicity for continuous processes defined on the time interval I = ℝ or I = [0, + ∞) with values in a separable Banach space 𝔼 (or more generally a separable completely regular topological space): almost periodicity in distribution, in probability, in quadratic mean, almost sure almost periodicity, almost equi-almost periodicity. In the deterministic case, all these notions reduce to Bochner-almost periodicity, which is equivalent to Bohr-almost periodicity when I = ℝ, and to asymptotic Bohr-almost periodicity when I = [0, + ∞).

Keywords:

• Almost periodic in distribution
• Almost periodic in p-mean
• Asymptotically almost periodic
• Bochner-almost periodic
• Bohr-almost periodic
• Continuous stochastic process

Mathematics Subject Classification:

• Primary 60G05
• Secondary 60H10, 34C27, 42A84

Acknowledgments

The authors heartily thank François Charlot for pointing outwith a counterexample (not included here) some errors in a previous version of this work. They are also grateful to Thierry de la Rue for Counterexample 2.16.