Proving $S_m=\sum\limits_{j=1}^m{\theta^j X_{n-j}}$ converges in mean square as $m\to \infty$

Question

Suppose that $\{X_t, t = 0, \pm1,\dots\}$ is is stationary and that $|\theta|<1$.

Show that for each fixed $n$, the sequence $S_m=\sum\limits_{j=1}^m{\theta^j X_{n-j}}$ converges in mean square as $m\to \infty$.

As I know, the sequence $S_n$ converges in mean square to some random variable if only if

$$E(S_k - S_h)^2 \xrightarrow{h,k\to \infty} 0$$

Assume $k > h$, then I have to prove $$ E\left( {\sum\limits_{j = h + 1}^k {{\theta ^j}{X_{n - j}}} } \right)^2 \xrightarrow{h,k\to \infty} 0 $$

Please give me a hint to finish this. Thank you.

Answer 1

Expand the square in order to get $$ \mathbb E\left( {\sum\limits_{j = h + 1}^k {{\theta ^j}{X_{n - j}}} } \right)^2 =\sum_{j=h+1}^k\sum_{j'=h+1}^k\theta^j\theta^{j'}\mathbb E\left[X_{n-j}X_{n-j'}\right]. $$ Then use stationarity and Cauchy-Schwarz inequality in order to bound $\left\lvert \mathbb E\left[X_{n-j}X_{n-j'}\right]\right\rvert$ by something independent of $n$, $j$ and $j'$.