What is the virtue of loading absolutely-summability in the definition of causality of ARMA model?

Question

An ARMA series $y_t$ is causal function of $\nu_t$ if there exists constants $\psi_j$ such that

$\sum_{j=0}^{\infty} |\psi_j|<\infty$ and $y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}<\infty$ for all $t\in \mathbb{Z}$.

In this definition, why do we load $\sum_{j=0}^{\infty} |\psi_j|<\infty$ (absolute-summability) to the definition?

Why is the ordinary summability $\sum_{j=0}^{\infty} \psi_j<\infty$ not preferred instead? What is the virtue of loading absolutely-summability, if any?

PS: Tags like summability, absolutely-summable would be useful in stats.stackexchange.

Answer 1

I don't know what you mean by "load".

You question is really about absolute-summability (i.e. $l^1$ condition) of MA$(\infty)$ coefficients. "Causal" just means it's a one-sided MA$(\infty)$ representation, but MA$(\infty)$ just the same.

Without $l^1$ condition, convolution of sequences does not one make sense in general. In the time-series context, this means the autocovariance function of a MA$(\infty)$ time series would not be defined, this is clearly problematic, in many ways.

For example:

1. Calculations that establish elementary LLN's and CLT's for covariance-stationary time series are basically mechanical manipulations of the autocovariance function. So no ACF, no inference.

2. Going one step further, the spectral density is the Fourier transform of ACF. No ACF, no discussion of spectral aspects of covariance-stationary time series.