Average of sequence random variables

Question

Let $X_1, X_2, X_3, \dots$ be a sequence of random variables that converges almost surely $$(X_n) \rightarrow X$$ to a number $X \in \mathbb R$ (or more precisely the delta dirac distribution centered at some real number). I am interested in the convergence of the average sequence $(Y_n)$ where $$Y_n = \frac 1 n \sum_{i=1}^n X_i$$ Can we make any statement about the convergence of $(Y_n)$? Intuitively, I would guess it also has to converge to $X$, but I am not sure about the mode and how to prove it.

Answer 1

For numerical sequences $(x_n)$, it is a standard exercise that $\lim x_n=L$ implies $\lim \frac{1}{n}\sum_{k=1}^n x_k=L$.

By assumption, there is a set $E$ of full measure such that for every $\omega\in E$, $X_n(\omega)\to X(\omega)$. By the above, this implies $Y_n(\omega)\to X(\omega)$.

Thus, $Y_n \to X$ almost surely. (And consequently, converges in probability and in distribution.) This is pretty much it; since there is no reason for $X$ or $Y_n$ to be integrable, we cannot expect, say, convergence in $L^1$.