[Math] Examples of convergence almost surely

Question

I know the definition of almost sure convergence, but I don't understand it in practice….for example, if I have independent random variables $$X_n \sim U (1, 1+ 1/n),$$ does it converge almost surely?

Thanks to all!

Answer 1

We usually use some auxilliary results to prove almost sure convergence. The Borel–Cantelli lemma is a very useful result that can be used to prove that a sequence of random variables converges almost surely. If we prove that $$ \sum_{n=1}^\infty\Pr\{|X_n-1|>\varepsilon\}<\infty $$ for each $\varepsilon>0$, then we have that the events $\{|X_n-1|>\varepsilon\}$ happen only a finite number of times almost surely (we use the Borel-Cantelli lemma here). This means that the sequence $\{X_n:n\ge1\}$ converges to $1$ almost surely.

Since $X_n\sim U(1,1+1/n)$, we have that $$ \Pr\{|X_n-1|>\varepsilon\}=\Pr\{X_n-1>\varepsilon\}=\Pr\{X_n>1+\varepsilon\}=\Pr\{X_n\in(1+\varepsilon,1+1/n)\}. $$ If we take sufficiently large $n$, the last probability above is $0$ and the series converges since only a finite number of terms are not equal to zero. Hence, $X_n$ converges to $1$ almost surely.