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?