# 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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### $X_1, X_2, \ldots$ are independent, $X_n \longrightarrow 0$ almost surely. Prove that $\sum_{n=1}^{\infty} P \left( |X_n > 1| \right)$ is convergent

Random variables $X_1, X_2, \ldots$ are independent, $X_n > \longrightarrow_{n\longrightarrow \infty} 0$ almost surely. Prove that $\sum_{n=1}^{\infty} P \left( |X_n > 1| \right)$ is convergent. ...
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### Question on application of Borel-Cantelli lemma

I have an exercise in my measure theory class which goes as following: If $(X,A,\mu)$, where $A$ is a $\sigma$-algebra, is a measure space and $(B_n),n=1,2,\ldots$ a sequence of sets in $A$. Show that ...
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### Borel Cantelli under dependence but conditionally bounded below by a positive probability

Let $(A_j)_{j\in\mathbb N}$ be a sequence of events. Think of them like a temporal sequence of coin flips. We will sequentially check to see if each $A_j$ occurs. Let events $E_j$ denote either $A_j$ ...
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### Borel-Cantelli lemma and countable intersection manipulation at the end of a complicated problem

I am struggling to understand how to apply Borel-Cantelli lemma at the end of a complicated problem. This is the problem: The expressions I am having trouble with are the last two probability ...
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### For each $M>0$, we have $\mathbb P(|X_n|>Mn \;\text{ i.o.})=1$

If $\{X_n\}$ is a sequence of iid random variables such that $\mathbb P(X_1=\pm j)=\frac c{j^2\log j}$, $j=3,4,\dots$, where $c$ is the appropriate normalizing constant. Show that for each $M>0$, ...
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### $X_n \to 0$ a.s. if and only if $\sum_n \mathbb{P}(X_n = 1) < \infty$.

This post is quite similar to the question "Prove that : $X_n \xrightarrow{\mathrm{a.s.}}0\iff \sum_n P(X_n>0) <\infty$", still slightly different. The statement is $X_n \to 0$ a.s. if ...
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### David Williams' Exercise 4.5 $\mathbb{P}(\text{limsup}(\frac{X_n}{\sqrt{2\log{n}}})\leq1)=1$

I am attempting Exercise 4.5 from David Williams' Probability with Martingales, which is about Borel-Cantelli lemma. The question states the follows. If $G$ is a random variable with the normal N(0,1) ...
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### Borel Cantelli-Type problem from Billingsley's Probability and Measure

Problem 4.11(d) from Billingsley's Probability and Measure book states: Show that $P\left(\limsup A_n\right)=1$ if and only if $\sum_n P(A\cap A_n)$ diverges for each $A$ of positive probability. ...
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### (convergence and probability) If $X_n$converges on the probability in $X$. Prove...

If $X_n$ converges on the probability in $X$. Prove a) (using only the definition of convergence with probability) For every $\epsilon_k \to 0$ when $k \to \infty$, that there exists a $n_k$ such ...
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### Kolmogorov Zero One Law

First Part :I want to prove that, if I have a sequence of independent Randomvariables $X_{n}$ and $T=\{\exists N \in \mathbb{N}:\forall n\geq N, X_{n}=X_{n+2}\}$, that $P(T)\in\{0,1\}$. I know that I ...
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I have the following problem. We have $Y_n$ independent random variables defined on $(\Omega, F, \Bbb{P})$ s.t. $$\Bbb{P}(Y_n=1)=p~~~\Bbb{P}(Y_n=0)=1-p$$ for $p\in [0,1]$. We define $A_0=0$ and $A_n=\... 2 votes 1 answer 157 views ### Sum of moments are finite, show that converg to zero. Consider$\{X_n\}$that for some$p>0$, the sum$ \sum_{i=0}^{\infty}{E|X_n|^p} $is finite. Show that$X_n\to 0$almost surely. Can someone help me find a way to solve this problem, or give a clue.... 2 votes 1 answer 68 views ### Almost Sure Convergence for dependent random variables Let's consider a sequence$X_1,X_2...$such that$P(X_n=0)=\frac{1}{n}$and$P(X_n=1)=1-\frac{1}{n}$. Prove that$\{X_n\}$doesn't converge, almost surely. So I have solved problem for Independent$\{...
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Let $X_1,X_2, \ldots$ r.v i.i.d. with geometric distribution and parameter $p = 1 - e^{-1}$, i.e. $P(X_1 = k) = p(1 -p)^k;$ $k = 0, 1, 2, \ldots$. Prove that P\left[\limsup_{n \to \infty}\frac{X_n}{...
Our professor for Introduction to Probability 1 course gave us the following question as an exercise: Let $X_i\sim Uniform[0,1.1]$ iid and let $Y_n=\prod_{i=1}^n X_i$. Does \$Y_n\overset{\text{a.s}}{\...