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

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.

• The causality tag is not really appropriate here. – Michael Aug 10 at 12:21
• $\sum_{j=0}^\infty \psi_j <\infty$ doesn't make much sense, since it allows the sum to diverge to $-\infty$. – gunes Aug 16 at 10:59
• @gunes That probability is almost always omitted. Written as that, it denotes the finiteness from both ends. – Erdogan CEVHER Aug 16 at 12:20

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.