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).
449
questions
2
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1
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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.
...
- 1,141
0
votes
0
answers
63
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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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1
vote
0
answers
47
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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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3
votes
1
answer
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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 $$\lim_{n\rightarrow\infty}\sqrt[n]{X_{1}\dots X_{...
- 5,575
2
votes
1
answer
62
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Relaxed Borel-Cantelli Lemma
Let ${p_1,p_2,\ldots \in [0,1]}$ be a sequence such that ${\sum_{n=1}^\infty p_n = +\infty}$. Show that there exist a sequence of events ${E_1,E_2,\dots}$ modeled by some probability space ${\Omega}$, ...
- 213
1
vote
0
answers
26
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Infinite Linear Combination of symmetric Random Variables
Let $(Y_n)$ be i.i.d. random variables taking values $1$ and $-1$ with equal probabilities. I want to compute the function
$$
f(x) = \mathbb{P}[\sum_n x_nY_n \text{ converges}]
$$
defined on sequences ...
- 429
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votes
0
answers
85
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What does Borel's large number theorem really mean?
According to this page in "Encyclopedia in Mathematics", the Borel's large number theorem can be stated as below.
"Consider independent random variables $X_1,\dots,X_n,\dots$ which are ...
- 43
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votes
0
answers
62
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Borel-Cantelli Lemma for poisson distribution
I'm working on the following excersise:
Given $x\in(0,\infty)$ and $0\le\lambda_n\lt x, \forall n\in\mathbb{N}$ and the sequence of random variables $(X_n)_{n \in \mathbb{N}} \sim Poi(\lambda_n)$ show ...
- 167
0
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1
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39
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Explanation of this passage (Borel Cantelli part 2)
I'm trying to understand a passage in the proof of Borel Cantelli Lemma part 2.
Be $(\Omega, \mathcal{F}, P)$ a probability space and $(A_n)$ a sequence in $\mathcal{F}$ such that $(A_n)$ are pair ...
- 2,696
1
vote
1
answer
52
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Clarification on the proof for Borel Cantelli lemma part $1$
I need some clarification about the proof for the BC lemma part $1$.
I am about to writing down the proof as it's written in my professor's notes and I assume BC lemma is known so I won't write it.
So ...
- 2,696
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votes
1
answer
46
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Clarification on Borel Cantelli theorem, part 1
I am having troubles in understanding a part of the proof of the part $1$ Borel Cantelli lemma.
Let $(\Omega, \mathcal{F}, P)$ a probability space and $(A_n)$ a sequence in $\mathcal{F}$.
$$\sum_{n = ...
- 2,696
2
votes
1
answer
70
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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 $\...
- 103
-2
votes
1
answer
51
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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 ∩...
0
votes
1
answer
70
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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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vote
1
answer
41
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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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2
votes
1
answer
70
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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 $...
- 423
1
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1
answer
118
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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 ...
- 78
0
votes
0
answers
43
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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, ...
- 159
3
votes
3
answers
63
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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 ...
- 2,696
0
votes
1
answer
32
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 ...
- 345
1
vote
1
answer
182
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 ...
- 583
0
votes
1
answer
81
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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 $$
P\left(\limsup _{n \rightarrow \infty}\left|X_n\right| / n^{1 / p} \leq \varepsilon\...
- 47
2
votes
2
answers
51
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Given $\{A_n\}$ with $P(A_n)\to 0$ show that for every $\alpha>0$ there exists a subsequence $\{A_{n_k}\}$, such that $P(\cup_k A_{n_k})\leq\alpha$
Given $\{A_n\}$ with $P(A_n)\to 0$ show that for every $\alpha>0$ there exists a subsequence $\{A_{n_k}\}$, such that
$$P(\cup_k A_{n_k})\leq\alpha$$
I found this variation of the problem I am ...
- 113
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votes
1
answer
171
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$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 ...
2
votes
1
answer
91
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$ ...
- 23
1
vote
0
answers
145
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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)$,...
- 309
2
votes
0
answers
72
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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^{...
- 431
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votes
1
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61
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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 ...
- 590
3
votes
1
answer
153
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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 ...
- 508
0
votes
1
answer
54
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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{...
user1115212
2
votes
0
answers
97
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\...
- 4,278
4
votes
0
answers
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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\...
- 4,278
1
vote
0
answers
54
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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 $\{...
- 2,400
2
votes
1
answer
235
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(...
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votes
1
answer
85
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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'...
- 2,400
4
votes
1
answer
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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 $E\text{ log}^+|X_1|<\infty \...
- 1,057
1
vote
1
answer
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$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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0
answers
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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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votes
1
answer
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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.
...
- 2,074
2
votes
1
answer
238
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Almost sure convergence by Borel-Cantelli
Assume that I have two sequences $(X_n)_{n \in \mathbb{N}}$ and $(Y_n)_{n \in \mathbb{N}}$ of random variables. Further assume $X_n \overset{\mathcal{D}}{=} Y_n$ and that $$\tag{$*$}\frac{X_n}{n} \to ...
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votes
1
answer
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If $E(X_n^2) = \infty$ then $\limsup \frac{|X_n|}{\sqrt{n}} \geq a$ almost surely.
We have given $X_1,X_2,\ldots$ an i.i.d. sequence of random variables such that $$\Bbb{E}(X_1^2)=\infty$$ I claim that for all $a>0$ $$\Bbb{P}\left(\limsup_{n\rightarrow \infty} \frac{|X_n|}{\sqrt{...
- 2,669
1
vote
1
answer
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Borel-Cantelli lemma implications
My previous question here on stack exchange has caused a follow-up question.
The Borel-Cantelli lemma tells us that, if for any $\varepsilon > 0$ we have
$\sum_{n=1}^\infty \mathbb{P}(\vert X_n - ...
- 159
3
votes
1
answer
121
views
(Borel-Cantelli) If $X_i$ is a sequence of identical and independent variables. Using Borel-Cantelli, prove that...
If $X_i$ is a sequence of identical and independent variables. Using Borel-Cantelli, prove that
$$E|X_1| \lt \infty \to P(|X_n| \gt n\ \text{infinite times}) = 0.$$
(uses that $E(x) = \sum_{n=0}^{\...
- 87
3
votes
1
answer
122
views
(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 ...
- 27
1
vote
2
answers
570
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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 ...
- 63
2
votes
1
answer
90
views
How do I rewrite this probability statement using $\limsup$?
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=\...
- 2,669
2
votes
1
answer
157
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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....
- 51
2
votes
1
answer
68
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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 $\{...
- 51
1
vote
2
answers
201
views
Geometric distribution and Borel Cantelli lemma
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}{...
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1
vote
1
answer
131
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Almost surely converges for product of Uniform[0,1.1] using defention
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}}{\...
- 77