# Defining Properly MA(∞) Time Series

In order to properly define series, we need to get back on some properties of infinite sequences, as briefly mentioned yesterday in the MAT8181 course. Consider some sequence . The sequence is said to be summable if

is convergent, i.e. if the limit of  exists when .

From Cauchy criterion converges if and only if for each , there is  for which

when . The sequence  is said to be absolutely summable if

and square-summable if

Observe that absolute summability will imply square summability (since for ‘s large enough , and then )

Consider now some  time series

If the sequence of coefficients  is square-summable, then

converges in   to some random varible as . This can be proved easily using Cauchy criteria, in the sense that for any , there is a  large enough such that, for any ,

In that case, if the sequence of coefficients  is square-summable, then  is stationary (in the  sense) since the process is centered, and

for all .

Further, ergodicity of the time series, define as the absolute summability of the autocovariance sequence, is obtained when the sequence of coefficients  is absolutely summable.

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