A note on the Berman condition

TL;DR

This paper demonstrates that certain conditions on time series, including the Berman condition, are preserved under filtering and ARMA modeling, ensuring their applicability in time series analysis.

Contribution

It establishes that the Berman and related summability conditions are maintained through filtering and invertible ARMA models, extending their theoretical applicability.

Findings

Berman condition is preserved under specific filters.

ARMA models maintain the Berman and summability conditions.

Results support the use of these conditions in time series analysis.

Abstract

It is established that if a time series satisfies the Berman condition, and another related (summability) condition, the result of filtering that series through a certain type of filter also satisfies the two conditions. In particular it follows that if $X_t$ satisfies the two conditions and if $X_t$ and $a_t$ are related by an invertible ARMA model, then the $a_t$ satisfy the two conditions.