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.