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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?

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    Sorry, what's a Plim?2011-04-30
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    $\mu_n = \mathbf{E}((\sum\lim_{i=1}^n (X_t-\bar X_t)^2)^{-1})$ is a *sequence of numbers*. There is no notion of convergence in probability in your statement.2011-04-30
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    Ewain: you could show the part of the proof you said you did, this could clarify the question you ask.2011-05-20
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    Traditionally (well, in my experience) the "autoregressive" processes are the stable ones (poles inside the unit circle). This would be more commonly called a random walk process, with continuous jumps.2011-07-19

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