Defining Properly MA(∞) Time Series

In order to properly define https://latex.codecogs.com/gif.latex?MA(\infty) series, we need to get back on some properties of infinite sequences, as briefly mentioned yesterday in the MAT8181 course. Consider some sequence https://latex.codecogs.com/gif.latex?(a_i)_{i\in\mathbb{N}}. The sequence is said to be summable if

https://latex.codecogs.com/gif.latex?S_n=\sum_{i=0}^n%20a_i

is convergent, i.e. if the limit of https://latex.codecogs.com/gif.latex?S_n exists when https://latex.codecogs.com/gif.latex?n\rightarrow\infty.

From Cauchy criterionhttps://latex.codecogs.com/gif.latex?\sum%20a_i converges if and only if for each https://latex.codecogs.com/gif.latex?\eta%3E0, there is https://latex.codecogs.com/gif.latex?m\in\mathbb{N} for which

https://latex.codecogs.com/gif.latex?\vert%20a_i+a_{i+1}+\cdots+a_{j-1}+a_j\vert%3C\eta

when https://latex.codecogs.com/gif.latex?%20i,j%3Em. The sequence https://latex.codecogs.com/gif.latex?(a_i)_{i\in\mathbb{N}} is said to be absolutely summable if

https://latex.codecogs.com/gif.latex?%20\sum_{i=0}^\infty%20\vert%20a_i\vert%20%3C\infty

and square-summable if

https://latex.codecogs.com/gif.latex?%20\sum_{i=0}^\infty%20a_i^2%20%3C\infty

Observe that absolute summability will imply square summability (since for https://latex.codecogs.com/gif.latex?j‘s large enough https://latex.codecogs.com/gif.latex?%20\vert%20a_j\vert%20\leq1, and then https://latex.codecogs.com/gif.latex?%20a_j^2\leq\vert%20a_j\vert)

Consider now some https://latex.codecogs.com/gif.latex?MA(\infty) time series

https://latex.codecogs.com/gif.latex?%20X_t=\sum_{h=0}^\infty%20\theta_h%20\varepsilon_{t-h}

If the sequence of coefficients https://latex.codecogs.com/gif.latex?%20(\theta_i) is square-summable, then

https://latex.codecogs.com/gif.latex?%20S_T%20=%20\sum_{h=0}^T%20\theta_h%20\varepsilon_{t-h}

converges in https://latex.codecogs.com/gif.latex?%20L_2  to some random varible as https://latex.codecogs.com/gif.latex?%20T\rightarrow\infty. This can be proved easily using Cauchy criteria, in the sense that for any https://latex.codecogs.com/gif.latex?\eta%3E0, there is a https://latex.codecogs.com/gif.latex?%20T large enough such that, for any https://latex.codecogs.com/gif.latex?%20h,

In that case, if the sequence of coefficients https://latex.codecogs.com/gif.latex?%20(\theta_i) is square-summable, then https://latex.codecogs.com/gif.latex?%20(X_t) is stationary (in the https://latex.codecogs.com/gif.latex?%20L_2 sense) since the process is centered, and

https://latex.codecogs.com/gif.latex?\gamma(h)=\sigma^2%20\cdot%20\sum_{i=0}^\infty%20\theta_i%20\theta_{i+h}

for all https://latex.codecogs.com/gif.latex?h\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 https://latex.codecogs.com/gif.latex?%20(\theta_i) is absolutely summable.


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