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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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A generalization of Borel-Cantelli's lemma

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $I_n = \bigcap_{k = 1}^{n}E_k$ so that $\{I_n\}_{n\in\mathbb{N}}$ is a decreasing sequence of events ($I_{n+1} \subseteq I_{n}$, $\forall n\...
MathRevenge's user avatar
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1 answer
33 views

Proving that $\liminf_{n\rightarrow\infty} \log X_n/\log n \le -1$ a.s. where $X_n$ are i.i.d. random variables, uniformly distributed on $[0,1]$

Let $X_n$, $n\in \mathbb{N}$ be independent random variables, uniformly distributed on $[0,1]$. Show that $$\liminf_{n\rightarrow\infty} \frac{\log X_n}{\log n} \le -1 \ \ \text{a.s.}$$ I added my ...
Ata Keskin's user avatar
0 votes
1 answer
61 views

Almost Sure Convergence/Strong Law for the Coupon Collector's Problem

Consider the usual coupon collector's problem - $n$ coupons with equal probability, and let $T_n$ be the time required to collect all $n$ coupons. It is well known that: $$ \mathbb{E}[T_n] = nH_n $$ ...
rubikscube09's user avatar
  • 3,915
1 vote
2 answers
164 views

Let $P(X_n=n)=1/n^a$ and zero otherwise. Let $Y_n=X_{n+1}\cdot X_n$. Find all $a$ such that $\liminf Y_n=0$ and $\limsup Y_n=\infty$ almost surely.

Remark: My attemps is WRONG as I fasly assumed that the $Y_n$ were independant. I ve corrected this in my own answer to this post. This answer is correct but not enough well wrotten, read the selected ...
OffHakhol's user avatar
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0 answers
22 views

Criterium for recurrence and Borel-Cantelli lemma

Given a Markov chain $X$ (discrete time, countable state space) we know that a state $i$ is recurrent iff $$\sum_{n\geq1}p_{ii}^{(n)}=\infty$$ where $p_{ii}^{(n)}=P(X_n=i|X_0=i)$. Otherwise, $i$ is ...
Francesco Virgili's user avatar
4 votes
2 answers
120 views

Second Borel-Cantelli lemma via the moment method

Let ${E_1,E_2,\dots}$ be a sequence of jointly independent events. If ${\sum_{n=1}^\infty {\bf P}(E_n) = \infty}$, show that almost surely an infinite number of the ${E_n}$ hold simultaneously. (Hint: ...
shark's user avatar
  • 1,011
-2 votes
1 answer
70 views

Deducement of First and second Borel-Cantelli Lemma

Suppose that $\Omega$ is a set, $(\Omega, \mathscr{G})$ is a measure space, and $Z: \Omega \to \mathbb{R}$ is a given mapping. Then Z is $\mathscr{G}$ measurable iff $$Z =\displaystyle\sum_{i=1}^\...
Win_odd Dhamnekar's user avatar
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Is it OK to say $\{ \sum_{n \geq 1} \mathbb{1}_{A_k} = \infty \} = \{ A_k \text{ i.o.}\}$

I have the following exercise: "For a sequence $(A_k)_{k \geq 1}$ of events, let $N(\omega) = \sum_{k \geq 1} \mathbb{1}_{A_k} ( \omega)$ be the number of $A_k$'s which occur. Show that if $\sum_{...
Ryderr's user avatar
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1 answer
51 views

Showing an independent sequence does not converge almost surely using Borel-Cantelli

I can't seem to write a solid proof for the following problem: Consider an independent sequence $\left(X_n\right)_{n \in \mathbb{N}}$ of random variables with $$ P\left(X_n=0\right)=1-\frac{1}{n \log (...
FinEng's user avatar
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1 vote
1 answer
45 views

Existence of sequence of Borel subsets of the unit interval

Consider the probability space $([0,1], \mathscr{B}([0,1]), \lambda)$, where $\lambda$ is the Lebesgue measure. Question. Does there exist a sequence $(S_n)$ of Borel subsets of $[0,1]$ with the ...
Paolo Leonetti's user avatar
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Proving that the sum of prime reciprocals diverges using Borel Canteloni, 2nd version

I'm looking on feedback on the following proof. Most importantly, did I make any fundamental errors in my reasoning? If not, how could I make the proof more professional, adding rigour and better ...
AndroidBeginner's user avatar
2 votes
1 answer
107 views

$X_1, X_2, \ldots$ 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. ...
fragileradius's user avatar
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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 ...
NoetherBoy 's user avatar
1 vote
0 answers
62 views

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 $A_j$ ...
jdods's user avatar
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3 votes
1 answer
110 views

If $(X_{i})$ are i.i.d nonnegative with $\mathbb{P}(X_{1}=0)>0$, show that $\lim \sqrt[n]{X_{1}\dots X_{n}}=0$ (surely).

I am working on an exercise as follows. Given i.i.d non-negative random variables $(X_{i})_{i=1}^{\infty}$ with $\mathbb{P}(X_{1}=0)>0$. Show that $$\lim_{n\rightarrow\infty}\sqrt[n]{X_{1}\dots X_{...
JacobsonRadical's user avatar
2 votes
1 answer
79 views

Relaxed Borel-Cantelli Lemma

Let ${p_1,p_2,\ldots \in [0,1]}$ be a sequence such that ${\sum_{n=1}^\infty p_n = +\infty}$. Show that there exist a sequence of events ${E_1,E_2,\dots}$ modeled by some probability space ${\Omega}$, ...
shark's user avatar
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1 vote
0 answers
35 views

Infinite Linear Combination of symmetric Random Variables

Let $(Y_n)$ be i.i.d. random variables taking values $1$ and $-1$ with equal probabilities. I want to compute the function $$ f(x) = \mathbb{P}[\sum_n x_nY_n \text{ converges}] $$ defined on sequences ...
Partial T's user avatar
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What does Borel's large number theorem really mean?

According to this page in "Encyclopedia in Mathematics", the Borel's large number theorem can be stated as below. "Consider independent random variables $X_1,\dots,X_n,\dots$ which are ...
Phil's user avatar
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0 answers
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Borel-Cantelli Lemma for poisson distribution

I'm working on the following excersise: Given $x\in(0,\infty)$ and $0\le\lambda_n\lt x, \forall n\in\mathbb{N}$ and the sequence of random variables $(X_n)_{n \in \mathbb{N}} \sim Poi(\lambda_n)$ show ...
Lost's user avatar
  • 157
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1 answer
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Explanation of this passage (Borel Cantelli part 2)

I'm trying to understand a passage in the proof of Borel Cantelli Lemma part 2. Be $(\Omega, \mathcal{F}, P)$ a probability space and $(A_n)$ a sequence in $\mathcal{F}$ such that $(A_n)$ are pair ...
Heidegger's user avatar
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1 vote
1 answer
95 views

Clarification on the proof for Borel Cantelli lemma part $1$

I need some clarification about the proof for the BC lemma part $1$. I am about to writing down the proof as it's written in my professor's notes and I assume BC lemma is known so I won't write it. So ...
Heidegger's user avatar
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0 votes
1 answer
47 views

Clarification on Borel Cantelli theorem, part 1

I am having troubles in understanding a part of the proof of the part $1$ Borel Cantelli lemma. Let $(\Omega, \mathcal{F}, P)$ a probability space and $(A_n)$ a sequence in $\mathcal{F}$. $$\sum_{n = ...
Heidegger's user avatar
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2 votes
1 answer
92 views

Application of Borel Cantelli lemma for series

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of real numbers, and let $(b_n)_{n \in \mathbb{N}}$ be a sequence of positive numbers. Show that if $\sum_{n \in \mathbb{N}}\sqrt{b_n}<\infty$ then $\...
Alex He's user avatar
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1 answer
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Proof for Statement regarding Borel Cantelli Lemma [closed]

I am struggeling to apply the Borel-Cantelli lamma to the following problem: Let $ A, A_1, A_2,\dots \in F $ in the probability space $ (\Omega, F, \textit{P}$) with $ \sum_{n \in \mathbb{N}} P ( A_n ∩...
Alligatooo's user avatar
0 votes
1 answer
95 views

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 ...
some_math_guy's user avatar
1 vote
1 answer
49 views

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 $M>0$, ...
Sayan Dutta's user avatar
  • 9,592
2 votes
1 answer
78 views

A paradox-like consequence of Borel-Cantelli's Lemma

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables, whose range is $\mathbb{N}$ (e.g., Geo(1)). Consider now the random sequence $X_1,X_2,X_3,...$ What is the probability that each $...
User271828's user avatar
1 vote
1 answer
231 views

Borel Cantelli Lemma Lim-sup Question

I am stuck on this one step in the proof of the first Borel-Cantelli Lemma. We have infinite $A_1, A_2, \ldots$ where the sum of their probabilities is finite. (convergence) Let $B$ be the event that ...
helixer's user avatar
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0 answers
59 views

Expected value of a series

While learning about the Borel-Cantelli Lemma, I came across this article and I am having trouble following the Introduction. I cite the relevant text here: Consider an infinite sequence of games, ...
ERed's user avatar
  • 197
3 votes
3 answers
73 views

Understanding this passage in Borel Cantelli Lemma N.2

I'm trying to understand a passage in the proof of Borel Cantelli Lemma 2. Be $(\Omega, \mathcal{F}, P)$ a probability space and $(A_n)$ a sequence in $\mathcal{F}$ such that $(A_n)$ are pair ...
Heidegger's user avatar
  • 3,482
0 votes
1 answer
41 views

Almost surely positive variables converging in law to an almost surely positive variable

I've been having some trouble with the following question: Let $X_1,X_2,...$ be almost surely positive random variables converging in distribution. Can you always find reals $(c_n)_{n\geq 1}$ such ...
Little Narwhal's user avatar
1 vote
1 answer
260 views

Borel Cantelli and convergence almost surely

I was wondering if it's possible to use the Borel Cantelli Theorem in order to ensure that the almost sure convergence DOESN'T exist. We know that: Let $A_n$ be a sequence of events in a probability ...
Ricter's user avatar
  • 583
0 votes
1 answer
105 views

limsup of random variables and Borel- Cantelli lemma

Let $X_n$ defined on a probability space for each $n\geq1$ The question is that $(X_n)_{n=1}^{\infty}$ is iid and $$ P\left(\limsup _{n \rightarrow \infty}\left|X_n\right| / n^{1 / p} \leq \varepsilon\...
Cloud's user avatar
  • 47
3 votes
2 answers
70 views

Given $\{A_n\}$ with $P(A_n)\to 0$ show that for every $\alpha>0$ there exists a subsequence $\{A_{n_k}\}$, such that $P(\cup_k A_{n_k})\leq\alpha$

Given $\{A_n\}$ with $P(A_n)\to 0$ show that for every $\alpha>0$ there exists a subsequence $\{A_{n_k}\}$, such that $$P(\cup_k A_{n_k})\leq\alpha$$ I found this variation of the problem I am ...
reaq-br4's user avatar
  • 123
2 votes
1 answer
283 views

$P(|X_n| \ge n \ i.o.) = 0$ if and only if $E[|X_1|] < \infty$.

Let $X_1,X_2,\dots$ be i.i.d. random variables. I want to show that $P(|X_n| \ge n \ i.o.) = 0$ if and only if $E[|X_1|] < \infty$. My thoughts: One direction is the straightforward application of ...
InsultedByMathematics's user avatar
2 votes
1 answer
124 views

After the $n$-th flip one gets $k$ consecutive Heads for a fair coin

Suppose one independently flips an infinite sequence of fair coins. Let $E_n$ be the event that the $n$-th coin is Heads. Let $A_n$ be the event that starting from the $n$-th flip one gets $k$ ...
Ayumi Bown's user avatar
1 vote
0 answers
158 views

Show that $\limsup_n \frac{S_n}{n(\log \log n)^{1-\epsilon}} = \infty$ with probability 1

Let $\{X_k\}$ be independent random variables such that $\mathbb{P}(X_k = k) = 1/k$ and $\mathbb{P}(X_k = 0) = 1-1/k$. Consider the sum $S_n = \sum_{k=1}^{n} X_k$. Show that for any $\epsilon\in(0,1)$,...
bdl10's user avatar
  • 309
2 votes
0 answers
78 views

Question on Borel Cantelli Lemma

Let $(\Omega,\mathbb{F},P)$ be a probability space and $\{X_n\}_{n=1}^{\infty}$ be a sequence of iid random variables such that for each $n \in \mathbb{N}$ and $r \in [1,\infty)$, $P(X_n > r) = r^{...
Jamal's user avatar
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0 votes
1 answer
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Borel Cantelli question [duplicate]

Let $X_1,X_2,\ldots$ be iid random variables. I want to show that $\mathbb{P}(|X_n|\geq n \text{ infinitely often})=0$ if and only if $\mathbb{E}(|X_1|)<\infty$. I think I can use Borel Cantelli ...
mathim1881's user avatar
2 votes
2 answers
351 views

a.s. convergence of Bernoulli sequence

Let $X_1, \ldots, X_n, \ldots$ be a sequence of Bernoulli random variables, $X_k \sim Bern(p_k)$. Prove that $$ X_n \xrightarrow{a.s.} 0 $$ if and only if $$ \sum_{k = 0}^{+\infty} p_k < +\infty. $$...
Paul's user avatar
  • 101
3 votes
1 answer
173 views

Central limit of independent indicator functions

Suppose $\{A_n\}_{n=1}^\infty$ is a sequence of independent events, each with probability $\mathbb{P}(A_n) = p_n$ such that $\sum_{n=1}^\infty p_n = \infty$. The goal here is to prove a stronger ...
mathmd's user avatar
  • 528
0 votes
1 answer
61 views

Version of Borel-Cantelli: $P(A_{i} \text{ i.o.)} > 0$

My fellow student stated the following after a lecture about the Borel-Cantelli lemma: Let $(A_{i})_{i \geq 0}$ be a sequence of disjoint of events on some probabilty space. If $\exists n \in \mathbb{...
user avatar
2 votes
0 answers
134 views

Show that $P\left(\left\{\liminf\limits_{n\to\infty}X_n\geq X\right\}\right)=1$

Let be $(\Omega,\mathcal{F},P)$ a probability space and $X_1,X_2,\dots$ a sequence of random variables. Show that if for a random variable $X$ and for all $\epsilon>0$ the infinite sum $\sum\...
Philipp's user avatar
  • 4,564
4 votes
0 answers
86 views

Show that $P\left(\limsup\limits_{n\to\infty}D_{m_n,i}\right)=1$ by means of Borel-Cantelli lemma

Let be $(\Omega,\mathcal{F},P)$ a probability space and $$ \limsup\limits_{n\to\infty} A_n:=\bigcap\limits_{n\geq 1}\bigcup\limits_{m=n}^{\infty}A_m,\text{ where } A_m\in\mathcal{F}\text{ for all } m\...
Philipp's user avatar
  • 4,564
1 vote
0 answers
69 views

Borel-Cantelli and almost sure convergence: Both events occur infintely often

Let $(X_n)_n$ a sequence of i.i.d random variables $\sim \operatorname{Ber}(1/n)$, i. e. $P(X=1)=1/n$ and $P(X=0)=1-1/n$. My question: Using Borel-Cantelli, I can show that both $\{X_n=1\}_n$ and $\{...
Analysis's user avatar
  • 2,482
2 votes
1 answer
401 views

For $X_1,\dots, X_n$ independent distributed with $P(X_n=0) = 1-\frac1n; P(X_n=n) = \frac1n$, does $X_n$ converge to $0$ almost surely?

I am referring to a question that has already been asked, but not completely answered (Why almost sure convergence holds if $X_n = n$ w.p. $1/n$?). For $X_1,\dots, X_n$ independent distributed with $P(...
BBB's user avatar
  • 45
2 votes
1 answer
111 views

On the Borel-Cantelli lemma - Is this argumentation valid?

In order to show that a $X_n$ converges to $X$ almost surely, one can often use the Borel-Cantelli lemma. I suppose it is a sufficient but not necessary condition for almost sure convergence. (?) Let'...
Analysis's user avatar
  • 2,482
4 votes
1 answer
328 views

Convergence of Random Power series

Q) Let $X_1,X_2,..$ be i.i.d. and not $\equiv 0$. Show that the radius of convergence of the power series $\sum_{n\geq 1}X_nz^n$ is $1$ a.s. or $0$ a.s. according as $E\text{ log}^+|X_1|<\infty \...
Dovahkiin's user avatar
  • 1,285
2 votes
1 answer
474 views

$X_n \to 0$ 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 ...
Vicky's user avatar
  • 931
2 votes
0 answers
227 views

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) ...
Chang's user avatar
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