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Suppose I have a series $X_t$ of random variables, $t \in \mathbb{N}_0$. I am not sure if the following reasoning is sound:

Let $f(x)$ be a function of the random variables.

Let $E[f(X_t)]$ denote the expectation value of $f$ for variable $t$, and let $E[f(X_t) | X_{t-1} = x]$ be the expectation value of $f(X_t)$ when we already know that $X_{t-1}$ had value $x$. Think of the $X_t$ as states of a system and $f(x)$ some function of these states.

I have proven the following result:

Lemma 1

If $f(x) > f_c$ for a certain critical value $f_c$, then $E[f(X_t) | X_{t-1} = x] \leq \alpha \cdot f(x)$ for $0 < \alpha < 1$.

I now want to prove the following:

Lemma 2

Let $T \ge 0$. Then either there is a $t < T$ so that $f(X_t) \leq f_c$, or it holds $E[f(X_T)] \leq \alpha^T \cdot E[f(X_0)].$

Proof
Either there is a $t < T$ so that $f(X_t) \leq f_c$. Then we are done. Or there is no such $t$ and we can use the previous bound: $E[f(X_T)] = E[E[f(X_T)|X_{T-1}=x]] \leq \alpha \cdot E[f(X_{T-1})]$ I can apply the induction hypothesis to that and obtain the claim.

Problem Now, I feel a bit queasy: In the expectation value, would I also have to define some event $\xi_T$ as the event that there is no $t < T$ such that $X_t \leq f_c$, and condition on that or is that, via the induction, already taken care of?

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    $X^T$ is a superscript and $\alpha^T$ is a power. I changed the sup- to subscripts to avoid this. That is the typical effect of "I have used that notation all the time so I intuitively now what it means, but know that someone points it out to me it really is ambigous" :)2011-05-05

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The conclusion that $E(X_T)\le\alpha^TE(X_0)$ for $T$ large enough cannot hold. Forget about probability for a minute and consider a deterministic sequence $(x_t)$ whose dynamics is $x_{t+1}=x_t+1$ if $x_t< x_c$ and $x_{t+1}=\alpha x_t$ if $x_t\ge x_c$. After a while, the sequence $(x_t)$ wil oscillate between $\alpha x_c$ and $x_c+1$ hence it cannot converge to zero.

Coming back to the probabilistic setting, the typical behaviour that your condition implies is that $(X_t)$ is positive recurrent and that the sequence $(E(X_t))$ converges to a positive and finite limit.

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    Or maybe real quick in the comments: If my condition in Lemma 2 would not read "Either there is a t < T such that $f(X_t) \leq f_c$", but rather it would read "Either there is a t < T such that $E[f(X_t)] \leq f_c$", could I then just prove this via induction on t?2011-05-05