Some Remarks on Definitions of Memory for Stationary Random Processes and Fields

Abstract

In the paper, we discuss how to define long, short, and negative memories for stationary processes on ℤ and fields on ℤ d with infinite variances. We propose to distinguish the properties of dependence and memory, and to attribute memory properties not only to a stationary random process (or a field) but also to a process and the operation that we apply to this process. We deal exclusively with the summation operation, that is, we consider the limit behavior of partial sums of random processes or fields. In order to have a unified approach to processes and fields with finite and infinite variances, we propose to define memory properties via the growth of normalizing sequences in limit theorems for partial sums. Also, we propose to change a little bit the terminology: instead of terms “long and short memories,” to use positive and zero memories, respectively, leaving the term “negative memory” and introducing “strongly negative memory.” For random fields, we introduce the notions of isotropic and directional memories.