Provide an example of a sequence of random variables which converge a.s. but not in mean.
I know that the random variables $X_n=n\cdot\mathbb{1}_{(0,\frac{1}{n})}$ converge in probability as given any $\varepsilon>0$
\begin{align*}
P(|X_n-0|>\varepsilon)=P(X_n>\varepsilon)\le P(X_n>0)=P\big(\big(0,\frac{1}{n}\big)\big)=\frac{1}{n}\to 0
\end{align*}
However, they do not converge in mean as
\begin{align*}
E|X_n-0|=E\big(n\cdot\mathbb{1}_{(0\frac{1}{n})}\big)=n\cdot P\big(\big(0,\frac{1}{n}\big)\big)=n\cdot\frac{1}{n}=1\,\,\text{for all n}
\end{align*}
So, my question here is do these $X_n$ converge to $0$ a.s.? And if so, how does one show this rigorously with an $\varepsilon$ proof? I know we need to find $N\in\mathbb{N}$ such that $X_N<\varepsilon$. To this end, we can make the lengths of the intervals $\big(0,\frac{1}{n}\big)$ arbitrarily small but the multiplication by $n$ stops $X_n$ from being arbitrarily small, so I am sort of thinking that these $X_n$ do not converge a.s. If thats the case, whats an example that would work here?
Best Answer
You want to prove that the event $\lbrace X_n(\omega) \rightarrow 0 \rbrace$ has probability $1$.
I claim that this event is the universe $\Omega$ itself. In fact, for any $\omega \in (0,1)$, the sequence $X_n(\omega)$ eventually vanishes.