Processes with block-associated increments

TL;DR

This paper introduces a new notion of association called block-association for processes with increments, characterizes it for Gaussian processes, and extends some CLT results to this setting.

Contribution

It proposes a natural new concept of block-association for vector-valued processes with increments and characterizes it for Gaussian processes using covariance supermodularity.

Findings

Block-association is equivalent to supermodularity of covariance functions in Gaussian processes.

The new notion generalizes association to processes with independent increments.

CLT for weak association extends to block-association.

Abstract

This paper is motivated by relations between association and independence of random variables. It is well-known that for real random variables independence implies association in the sense of Esary, Proschan and Walkup, while for random vectors this simple relationship breaks. We modify the notion of association in such a way that any vector-valued process with independent increments has also associated increments in the new sense --- association between blocks. The new notion is quite natural and admits nice characterization for some classes of processes. In particular, using the covariance interpolation formula due to Houdr\'{e}, P\'{e}rez-Abreu and Surgailis, we show that within the class of multidimensional Gaussian processes block-association of increments is equivalent to supermodularity (in time) of the covariance functions. We define also corresponding versions of weak association, positive association and negative association. It turns out that the Central Limit Theorem for weakly associated random vectors due to Burton, Dabrowski and Dehling remains valid, if the weak association is relaxed to the weak association between blocks.