I am trying to prove the following:
Let $X_t$ be a sequence of random variables that follows an autoregressive process; i.e. $X_t=X_{t-1}+e_t$, where $e_t$ is a zero mean i.i.d. sequence. Then $\lim\limits_{n\to \infty} \mathbf E\left(\left(\sum\limits_{i=1}^n (X_t-\bar X_t)^2\right)^{-1}\right)$ converges in probability to 0, where $\bar X_t$ denotes the sample mean.
So far, I have been able to show that $\mathbf E\left(\sum(X_t-\bar X_t)^2\right)$ converges in probability to infinity, but this alone does not imply the expectation of its reciprocal goes to 0. Are there any results I can use, along with this, to complete my proof?