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.