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.