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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 ifis convergent, i.e. if the limit of  exists when .From Cauchy criterion,  converges if and only if for each , there is  for whichwhen . The sequence  is said to be absolutely summable ifand square-summable ifObserve that absolute summability will imply square summability (since for 's large enough , and then )Consider now some  time seriesIf the sequence of coefficients  is square-summable, thenconverges 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, andfor 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.