# Questions tagged [borel-cantelli-lemmas]

For questions involving the Borel-Cantelli lemma or the second Borel-Cantelli lemma. Use this tag along with (probability-theory), (real-analysis) or (measure-theory).

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### Convergence almost surely question

Suppose you have a sequence of independent random variables {$X_i, i\geq1$}, such that $$\Bbb P(X_i=i^2 -1)=i^{-2},$$ $$\Bbb P(X_i=-1)=1-i^{-2}.$$ Then, $\frac1n \sum_{i=1}^n X_i$ converges almost ...
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### $X$ RV $X(S)=s$. Prove $X_n\underset{a.s}{\to}X$.

Let $([0,1],\mathcal{B}_{[0,1]},\lambda)$ probabilty space and let $X_n(s)=\frac{n}{n+1}s+(1-s)^n$ a sequence for RVs for every $s\in[0,1]$. and let $X$ RV $X(S)=s$. Prove $X_n\underset{a.s}{\to}X$. ...
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### >Let $X_1,\ldots,X_n$ indepent variable RV $X_n\sim Bern(1/n)$ Does the $X_n\underset{a.s}{\to}0$

Let $X_1,\ldots,X_n$ indepent variable RV $X_n\sim Bern(1/n)$ Does the $X_n\underset{a.s}{\to}0$ I tried to use Borel-Cantelli but I get that $\sum_{i=1}^{\infty}\mathbb{P}(A_n^{\epsilon})=\infty$ ...
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### Clarification on convergence almost surely

The three aspects within convergence in distributions include: Convergence in distribution Convergence in probability Convergence almost surely Convergence in distribution and in probability ...