# 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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### Borel-Cantelli argument for maximum of random variables

Let $(X_n)_{n \geq 1}$ be a sequence of random variables taking non-integer values, such that for each $n$ and each $i$, $\mathbb{P}(X_n \geq i) = 1/i$. By Borel-Cantelli, I have managed to show that (...
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### Infinite coin tosses- same number of heads and tails will occur infinitely many times

Dear math stack exchange community, It's my first post, so hello and please let me know if I haven't posted this question in a proper way- happy to edit it. So here's an exercise which I'm trying to ...
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### Convergence almost surely and Borel-Cantelli's Lemma (proof clarification)

I believe it is easier if I print the proof below: Several other papers uses the same argument to bound $\lvert R_n(x)-E R_n(x) \rvert$ almost surely, so I believe it is correct. It should be ...
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### Almost sure convergence of average of random variables

In my statistical inference course exercise guide, I am confronted with the following problem: Let $0<\theta<1/2$, and define the sequence $\{X_n\}_{n\in\mathbb{N}}$ of discrete independent ...
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### $\frac{X_n}{n}$ does not converge to $0$ almost surely

Suppose that $\sigma_{n}^{2} \geq 0, n \geq 1,$ satisfy $\sum_{n=1}^{\infty} \frac{\sigma_{n}^{2}}{n^{2}}=\infty$ and without loss of generality that $\sigma_{n}^{2} \leq n^{2}$ for all $n \geq 1 .$ ...
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### Proof of strengthening of Borel-Cantelli Lemma

If $A_1$,$A_2$, $\dots$ are pairwise-independent events with $\sum_{n \in \Bbb N} \Bbb P(A_{n})=\infty$ , then show for$$Q_{n}=\frac{\sum _{j=1}^{n} \Bbb 1_{A_{j}}}{\sum _{j=1}^{n} \Bbb P(A_{j})}$$ we ...
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### Convergence of series in probability to 1 implies convergence of pointwise max over $n$ in probability to 0

If we have non-negative random variables $X_k \geq 0$ such that $\frac{1}{n}\sum_{i = 1}^n X_i \to 1$ in probability. How do we show that $\frac{\max_{1 \leq k \leq n} X_k}{n} \to 0$ in probability? ...
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### Does $\text{Binomial}(2,1/n)$ converge to zero almost surely?

If I define $X_n=\text{Binomial}(2,1/n)$ then $P(X_n>\epsilon)=P(\text{Binomial}(2,1/n)>0)=1-P(\text{Binomial}(2,1/n)=0)=1-(1-1/n)^2$ Clearly $X_n$ converges to $0$ in probability But I can't ...
Let $\{Z_n\}_{n=1}^\infty$ be a sequence of i.i.d standard random variables. Prove that $P(\limsup_n |Z_n|/\sqrt{2ln(n)}=1)=1$. (Attempt) It is clear that both B.C lemmas will be used. Let \$\delta>...