A compact LIL for martingales in $2$-smooth Banach spaces with applications

TL;DR

This paper establishes a compact law of the iterated logarithm for martingales in 2-smooth Banach spaces, deriving invariance principles and strong laws for stationary processes, with novel results even in real-valued cases.

Contribution

It introduces new compact LIL results for Banach space-valued martingales and stationary processes, extending classical probabilistic laws to broader settings.

Findings

Proved compact LIL for Banach space martingales.

Derived almost sure invariance principles for stationary processes.

Established strong laws of large numbers in Banach spaces.

Abstract

We prove the compact law of the iterated logarithm for stationary and ergodic differences of (reverse or not) martingales taking values in a separable $2$-smooth Banach space (for instance a Hilbert space). Then, in the martingale case, the almost sure invariance principle is derived from a result of Berger. From those results, we deduce the almost sure invariance principle for stationary processes under the Hannan condition and the compact law of the iterated logarithm for stationary processes arising from non-invertible dynamical systems. Those results for stationary processes are new, even in the real valued case. We also obtain the Marcinkiewicz-Zygmund strong law of large numbers for stationary processes with values in some smooth Banach spaces. Applications to several situations are given.