GIVEN DATA: For a Bernouilli stochastic process $X_i$, let’s define:
\begin{equation*} Y_i=2X_i-1= \left\{ \begin{array}{ll} 1 & X_i=1 \\ -1 & X_i=0 \\ \end{array} \right. \end{equation*}
For each $n$,
$ S_n=\sum_{i=1}^{n}{Y_i}$, $S_0=0$
QUESTION: Show that whether or not the given random walk is mean ergodic.
MY WORKING:
I know that the sequence of random variables $S_0,S_1,S_2,...$ is a one dimensional random walk.
Also, any process $X(t)$ is mean-ergodic if the time average estime converges to the ensemble average $\mu_X$ as $T\to \infty.$
I don't know how do I use the above information to reach the final resul which the question asks me to show. i.e: whether or not the random walk is mean ergodic.
Does anyone have any idea how do I show it? Any guidance will be appreciated.
Thanks.
In my comment about the wide-sense stationarity of $\ S_n\ $ I confused the two indices. For $\ S_n\ $ to be wide-sense stationary, the autocovariance, $\ \mathbb{Cov}\big(S_nS_{n+r}\big)\ $, must be a function of $\ r\ $ alone, independent of $\ n\ $.
Let $$ p=\mathbb{P}\big(X_i=1\big)\ . $$ Then \begin{align} \mathbb{E}\big(Y_i\big)&=2p-1\ \text{, and}\\ \mathbb{Cov}\big(Y_i,Y_j\big)&=\mathbb{E}\left(\big(Y_i-(2p-1)\big)\big(Y_j-(2p-1)\big)\right)\\ &=4p(1-p)\delta_{ij}\ . \end{align} Therefore, \begin{align} \mathbb{Cov}\big(S_nS_{n+r}\big)&=\mathbb{E}\left(\sum_{i=1}^n\sum_{j=1}^{n+r}\left(\big(Y_i-(2p-1)\big)\big(Y_j-(2p-1)\big)\right)\right)\\ &=4p(1-p)\sum_{i=1}^n\sum_{j=1}^{n+r}\delta_{ij}\\ &=4p(1-p)\sum_{i=1}^n1\\ &=4p(1-p)n\ . \end{align} Since this is not independent of $\ n\ $, then $\ S_n\ $ is not wide-sense stationary, and the definition of mean ergodicity isn't applicable to it