Defining Properly MA(∞) Time Series

In order to properly define\infty) series, we need to get back on some properties of infinite sequences, as briefly mentioned yesterday in the MAT8181 course. Consider some sequence{i\in\mathbb{N}}. The sequence is said to be summable if\sum_{i=0}^n%20a_i

is convergent, i.e. if the limit of exists when\rightarrow\infty.

From Cauchy criterion\sum%20a_i converges if and only if for each\eta%3E0, there is\in\mathbb{N} for which\vert%20a_i+a_{i+1}+\cdots+a_{j-1}+a_j\vert%3C\eta

when,j%3Em. The sequence{i\in\mathbb{N}} is said to be absolutely summable if\sum_{i=0}^\infty%20\vert%20a_i\vert%20%3C\infty

and square-summable if\sum_{i=0}^\infty%20a_i^2%20%3C\infty

Observe that absolute summability will imply square summability (since for‘s large enough\vert%20a_j\vert%20\leq1, and then^2\leq\vert%20a_j\vert)

Consider now some\infty) time series\sum_{h=0}^\infty%20\theta_h%20\varepsilon_{t-h}

If the sequence of coefficients\theta_i) is square-summable, then\sum_{h=0}^T%20\theta_h%20\varepsilon_{t-h}

converges in  to some random varible as\rightarrow\infty. This can be proved easily using Cauchy criteria, in the sense that for any\eta%3E0, there is a large enough such that, for any,

In that case, if the sequence of coefficients\theta_i) is square-summable, then is stationary (in the sense) since the process is centered, and\gamma(h)=\sigma^2%20\cdot%20\sum_{i=0}^\infty%20\theta_i%20\theta_{i+h}

for all\in\mathbb{N}.

Further, ergodicity of the time series, define as the absolute summability of the autocovariance sequence, is obtained when the sequence of coefficients\theta_i) is absolutely summable.

Leave a Reply

Your email address will not be published. Required fields are marked *