Modulus of continuity for a martingale sequence

TL;DR

This paper proves that for bounded martingale sequences of random fields, the pointwise limit can be modified to preserve a given modulus of continuity, with additional smoothness and convergence properties under stronger conditions.

Contribution

It establishes conditions under which the modulus of continuity and smoothness are preserved in the limit of a martingale sequence of random fields.

Findings

The pointwise limit can be modified to preserve the modulus of continuity.

Stronger boundedness and smoothness lead to improved regularity of the limit.

Modulus of continuity is preserved for derivatives of the limiting field.

Abstract

Given a martingale sequence of random fields that satisfies a natural assumption of boundedness, it is shown that the pointwise limit of this sequence can be modified in such a way that a certain class of moduli of continuity is preserved. That is, if every element of the sequence admits a given modulus of continuity, one can construct a modification of the limiting random field so that this new field also admits the same modulus of continuity. Additionally, it is shown that requiring further smoothness and a stronger notion of boundedness for the original sequence guarantees further smoothness of the limiting field and a stronger mode of convergence to this limit. Moreover, the modulus of continuity is also preserved for the derivatives.