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.
OpenEdition suggests that you cite this post as follows:
Arthur Charpentier (January 28, 2014). Defining Properly MA(∞) Time Series. Freakonometrics. Retrieved December 3, 2024 from https://doi.org/10.58079/outv