Applying dominated convergence theorem on sum of random variables and indicator Consider i.i.d random variables $(X_k) \in \mathbb{N}$ and Y$_n=\sum_{i=1}^n X_i$ and $\mu= E(X_i)<1 $
Consider $E[(1-\frac{Y_n}{n} )1_{\{k \leq Y_n \leq n\}}]$ with $k \in [0, n]$
How can I apply dominated convergence to conculde that this expression goes to $1- \mu$
Why do I need the indicator here?
 A: Ok, since the $X_n$ are taking values in $\mathbb{N}$ we conclude that $0<\mu<1$.
The strong law of large numbers tells us that $1-\frac{Y_n}{n}\to 1-\mu$ almost surely. Also, let's take a look at the event $\{k\leq Y_n\leq n\}=\{\frac{k}{n}\leq \frac{Y_n}{n}\leq 1\}$. If $\omega$ is some point where $\frac{Y_n(\omega)}{n}\to \mu$ (from the strong law it happens at almost every $\omega$) then the condition $0<\mu<1$ implies that we eventually have $\frac{k}{n}\leq\frac{Y_n(\omega)}{n}\leq 1$, or equivalently $1_{\{k\leq Y_n\leq n\}}(\omega)=1$ for every large enough $n$. So for such $\omega$ we have $1_{\{k\leq Y_n\leq n\}}\to 1$. We conclude that:
$(1-\frac{Y_n}{n})1_{\{k\leq Y_n\leq n\}}\to 1-\mu$ almost surely
Also, note that for every $n$ we have:
$0\leq(1-\frac{Y_n}{n})1_{\{k\leq Y_n\leq n\}}\leq (1-\frac{k}{n})\leq 1$
So this allows us to use the dominated convergence theorem. We wouldn't be able to get that bound without the indicator. We indeed have $1-\frac{Y_n}{n}\to 1-\mu$ almost surely, but without the indicator we wouldn't be able to find a uniform bound on the sequence to use DCT.
