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.

  • $\begingroup$ The causality tag is not really appropriate here. $\endgroup$
    – Michael
    Aug 10, 2019 at 12:21
  • $\begingroup$ $\sum_{j=0}^\infty \psi_j <\infty$ doesn't make much sense, since it allows the sum to diverge to $-\infty$. $\endgroup$
    – gunes
    Aug 16, 2019 at 10:59
  • $\begingroup$ @gunes That probability is almost always omitted. Written as that, it denotes the finiteness from both ends. $\endgroup$ Aug 16, 2019 at 12:20

1 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.


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