Holderian weak invariance principle under a Hannan type condition

TL;DR

This paper studies the invariance principle in H{"o}lder spaces for stationary martingale differences, establishing new conditions on conditional variance that extend classical results and relate to Hannan's condition.

Contribution

It introduces a sufficient condition on conditional variance for the invariance principle in H{"o}lder spaces, extending Hannan's condition to stationary ergodic martingale differences.

Findings

The tail condition for i.i.d. sequences does not hold for stationary ergodic martingale differences.

A new sufficient condition on conditional variance guarantees the invariance principle.

The results connect Hannan's condition with invariance principles in H{"o}lder spaces.

Abstract

We investigate the invariance principle in H{\"o}lder spaces for strictly stationary martingale difference sequences. In particular, we show that the sufficient condition on the tail in the i.i.d. case does not extend to stationary ergodic martingale differences. We provide a sufficient condition on the conditional variance which guarantee the invariance principle in H{\"o}lder spaces. We then deduce a condition in the spirit of Hannan one.