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 
,
^2}_{\sigma^2%20\cdot%20\sum_{i=T}^h%20\theta_i^2}%20\leq%20\eta)
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.
