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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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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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{...
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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 - ...
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5 votes
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(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}^{\...
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1 answer
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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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1 answer
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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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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=\...
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34 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....
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2 votes
1 answer
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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 $\{...
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1 vote
2 answers
80 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
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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}}{\...
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3 answers
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Applications of Fatou's lemma and Borel-Cantelli lemma

Let $E_1, E_2, \cdots$ be a sequence of events such that $\inf_n \mathbb{P}(E_n)>0$. Prove that $\mathbb{P}\left(\sum_n 1_{E_n}=\infty\right)>0$. i.e. there is a positive probability that an ...
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1 answer
190 views

Proving $\frac{1}{n} \sum_{k=1}^{n}X_{k}\to 0$ a.s.

$\{X_{n}\}$ is a sequence of independent random variables, $EX_n=0$, and $\sum_{n=1}^{\infty}n^{-(r+1)}E(|X_n|^{2r})<\infty$. Proving $\frac{1}{n} \sum_{k=1}^{n}X_{k}\to 0$ a.s. and $r>1$ I ...
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1 vote
1 answer
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Using Borel-Cantelli to find limsup

Suppose $Y_1, Y_2, \dots$ is any sequence of iid real valued random variables with $E(Y_1)=\infty$ . Show that, almost surely, $\limsup_n (|Y_n|/n)=\infty$ and $\limsup_n (|Y_1+...+Y_n|/n)=\infty$. I ...
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1 answer
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How can we show a sequence convergence to almost surely in conditional expectation

Consider a sequence of i.i.d. random variables $\left\{Z_i\right\}$ where $\mathbb{P}\left(Z_i = 0\right) = \mathbb{P}\left(Z_i = 1\right) = \frac{1}{2}$. Using this sequence, define a new sequence of ...
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1 vote
2 answers
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Application of Borel Cantelli lemma 2

Suppose that a monkey sits in front of a computer and starts hammering keys randomly on the keyboard. Show that the famous Shakespeare monologue starting "All the worlds a stage" will ...
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3 votes
1 answer
132 views

Polya's urn as a counterexample for the Kolmogorov 0-1 law

Consider a simple formulation for the Polya urn model. An urn contains two balls at time 0, one is white and the other is black. At time $n\in\mathbb{N}$, one of the balls is chosen uniformly at ...
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2 votes
1 answer
93 views

Showing that a martingale $Y_k$ does not converge almost surely.

Let $X_i$ be iid with $$\mathbb{P}(X_i=1)= \mathbb{P}(X_i= -1) = \frac{1}{2i}, \mathbb{P}(X_i=0)=1-\frac{1}{i},$$ where $i=1,2,...$ And define $Y_1=X_1$ and for $k\geq2$ $$Y_k= \begin{cases} X_k, \...
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4 votes
1 answer
57 views

Almost sure convergence of rescaled nondecreasing sequences of random variables

Let us consider a sequence $(S_n)_n$ of $L^2$ random variables. Assume: $S_n \le S_{n+1}$ almost surely $S_n \to_{n \to \infty} +\infty$ almost surely $\frac{S_n}{\mathbb{E}[S_n]} \to_{n \to \infty} ...
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4 votes
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limsup of a sequence of subadditive random variables with subgaussian tail.

Suppose that $X_n$ is a sequence of random variables satisfying $\bullet P(X_n > t\sqrt{n}) < e^{-t^2}$, and $\bullet X_{m+n} < X_m+X_n$ for all $n,m$. We want to try and show that $$\limsup_{...
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1 answer
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A condition for long run boundedness for a stochastic process

Suppose we have a general continuous stochastic process $X_t$ which satisfies $$\lim_{m \rightarrow \infty} \sup_{t \ge 0} P \left( X_t < \frac{1}{m} \right) = 0 \quad \quad \quad \textbf{(1)}$$ I ...
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2 votes
1 answer
91 views

Is it necessarily true that $\sum\limits_{N \in \mathbb N}\sum\limits_{q \in \mathbb Q_{+}}\mathbb P(X>\frac{N-aq}{\sqrt{q}})<\infty$

Is it necessarily true that: $$\sum\limits_{N \in \mathbb N}\sum\limits_{q \in \mathbb Q_{+}}\mathbb P(X>\frac{N-aq}{\sqrt{q}})<\infty (*)$$ where $X$ is a standard normal distribution. The ...
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0 answers
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Given $X_1, X_2, ...$ are independent random variables uniform on $[-a_1, a_1], [-a_2, a_2], ...$, show that $\prod_{i=1}^n X_i \rightarrow 0$ a.s.

Given $X_1, ...$ defined as above for some sequence of $a_n>0$, and let $Y_n = \prod_{i=1}^n X_i$, show $Y_n \stackrel{a.s}{\rightarrow} 0$ if $\limsup_{n \rightarrow \infty} a_n < 2$. Here's my ...
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2 answers
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An application of the Borel-Cantelli lemma

Problem. Consider the independent standard Cauchy variables $ X_1, X_2, X_3, \ldots $, i.e. their probability density function is of form $f(x) = \dfrac{C}{1+x^2}$ (for some $C\in\mathbb{R}$). If $ Z =...
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1 vote
1 answer
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Is it possible that $P(\lim \sup \{X_n = 0\}) = P(\lim \sup \{X_n = 1\}) = 1$?

Perhaps this is a very silly question, and I'm probably just confusing some really obvious property, but suppose that I have a sequence of independent Bernoulli random variables with probability $1/n$....
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1 vote
1 answer
136 views

Understanding the concept of infinitely often

My book in stochastic processes has this section about i.o which I don’t really understand. Is there any example of any other to explain this? How can I understand this? It states the following: Let $...
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1 vote
1 answer
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Bound for series of probabilities of events

I am attempting to prove that a sequence of events occurs infinitely often with probability 0, similarly to Borel-Cantelli. I know the series of probabilities of set differences converges. This is ...
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0 votes
2 answers
41 views

Covering a probability space with a sequence of sets

Let $E_n$ be a sequence of events with $\inf_n {\bf P}(E_n) = \delta > 0$. I need to demonstrate that the lower bound $$\mathop{\bf P}\left( \sum_{n \leq N} 1_{E_n} \geq \delta \frac{N}{2} \right) \...
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0 votes
1 answer
205 views

Borel-Cantelli-Lemma and almost sure convergence

Suppose it holds: $P(|S_n|> \varepsilon )\leq \frac{1}{2^n \varepsilon^2} $, where $S_n $ is sequence of random variables. Furthermore it holds: $\sum_{n \geq 1} P[|S_n|> \varepsilon ] <\...
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  • 57
0 votes
1 answer
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Application of Borel cantelli lemma to a sequence of processes

Let $g^k$ be a sequence of processes satisfying $$P\left(\int_0^T |g^k-g^{k+1}|^2\,ds>2^{-k}\right) \leq 2^{-k}$$ Then the book I am reading states that as a application of the Borel-Cantelli lemma ...
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2 votes
0 answers
51 views

vertex percolation in an infinite graph

Suppose that $G=(V,E)$ is an infinite graph with finite degrees where every vertex is black with probability $p$. The black subgraph is a subgraph whose vertices are black and the edges are edges ...
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5 votes
0 answers
196 views

Probability a ball will stay in urn forever

Suppose that there is an urn with infinite capacity. In the first day, we put one ball in the urn and remove it. In the second day, we put 2 balls and remove one randomly. in the $k^{th}$-day, we put $...
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2 votes
0 answers
70 views

Prove $\sum_{i=1}^n X_i \to -\infty$ provided $P(X_k=k^2)=1/k^2$

Suppose $\{X_n,\,n\geq 1\}$ are independent with $$ \mathbb{P}(X_k=k^2) = \frac{1}{k^2},\quad \mathbb{P}(X_k=-1)=1-\frac{1}{k^2}. $$ Show that $$\sum_{i=1}^n X_i \to -\infty\quad \text{a.s.}$$ We are ...
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2 votes
1 answer
222 views

Intuition for Borel-Cantelli lemma and almost surely convergence

I have looked around the site to see if a similar question was asked. I couldn't find one, but please refer to one, if it happens I'm mistaken. My question is with regard to some intuition about the ...
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4 votes
0 answers
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almost sure convergency of $W_{n}=\frac{Z_{n}}{m^{n}}\to W$ implies $\frac{Z_{n}}{\mu^{n}}\to0$, where $\mu>m$.

I though to apply Borel-Cantelli, however it does not seem to work. m and $\mu$ are constants, with $0<m<\mu$, with random identically distributed variables $Z_{n}$, which take values in $\...
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1 vote
0 answers
33 views

Let $N_n$ be the length of the success run beginning at the $n$th trial. Prove that with probability one $\lim \sup\ \frac{N_n}{\log n} = 1$

($p = $ the probability of success, $q = 1-p$ and $\log$ denotes the logarithm of basis $1/p$) This problem comes from Feller's intro to probability and as a hint it suggest the following: "...
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3 votes
2 answers
184 views

A problem about the Borel-Cantelli lemma in Feller's Introduction to Probability

The problem is from Chapter 8 of the book, and it states the following. "In a sequence of Bernoulli trials let $A_n$ be the event that a run of $n$ consecuitive successes ocurrs between the $2^n$...
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  • 123
0 votes
1 answer
49 views

A series for which probability of infinitely many events is 0.

Given real-valued random variables $X_i$ for $i \geq 0$, I need to find a series of constants $a_i \in \mathbb{R}$ such that: $$P(X_i > a_i \text{ for infinitely many } i) = 0$$ My attempt is to ...
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1 vote
1 answer
56 views

Almost sure convergence of AR(1) model

I am trying to solve following problem. Problem. Suppose that $X_n = \rho X_{n-1} + \epsilon_n$ with $|\rho| < 1$ and $X_0 = 0$, where $\epsilon_n$ are iid r.v.'s with mean $0$ and variance $1$. ...
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4 votes
1 answer
108 views

Limsup of random variables vs limsup of events

In 2.7 of this notes, it is shown that, using Borel-Cantelli lemma, if $E_n = \left\{\frac{X_n}{\log n} \geq 1\right\}$ then $\mathbb{P}(\limsup \ E_n) = 1$ but why does author conclude that if $Y = \...
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3 votes
1 answer
80 views

Existence of a number using Borel-Cantelli

Let $\{a_n\}_{n=1}^\infty$ be a sequence of real numbers in $[0,1]$ and let $\{b_n\}_{n=1}^\infty$ be a sequence of positive real numbers such that $\sum_{n=1}^\infty b_n<\infty$. Show that there ...
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0 votes
1 answer
59 views

Sequence of random variable does not converge almost surely (but in P ?)

Let $(X_n)_{n \geq 1}$ be a sequence of iid random variables such that $\mathbb P(X_n = 0) = \frac{1}{2}$ and $\mathbb P(X_n = 1) = \frac{1}{2}$. For all $n \geq 1$, let $S_n := \sum_{k=1}^n 2^{k-n-1}...
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2 votes
0 answers
418 views

Application of Borel Cantelli lemma (almost sure convergence)

Let $(X_n)_{n\geq 1}$ be a sequence of real-valued random variables. I have to proof that if for every $\epsilon > 0: \sum_{n=1}^{\infty} \mathbb P(|X_n - X| > \epsilon) < \infty$ , then $X_n ...
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2 votes
1 answer
50 views

Problem involving Borel-Cantelli Lemma and Lim Sup of Sequences

Let $(\Omega,\mathcal{F},P)$ be a probability space, and $A_n$, $B_n$ two sequences of elements in $\mathcal{F}$ such that, $\sum\limits_{n=1}^{\infty}P(A_n\cap B_{n+1}^c)<+\infty$ and, $\sum\...
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  • 21
2 votes
1 answer
91 views

Proof of Borel-Cantelli Lemma explanation

I am trying to follow the proof of the Borel-Cantelli lemma as shown below: Could you please explain me how to go from: Thus $\sum \limits_{n = 1}^{\infty} 1_{A_n}$ is almost surely finite to: ...
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1 vote
1 answer
59 views

Borel-Cantelli Lemma for Poisson random variables

I want to solve a simple problem: I have a sequence of independent random variables $X_n$ distributed according to Poisson with expectation $\mathbb{E}_{X_n} = 1$ I want to prove that $\mathbb{P}\{\...
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1 vote
1 answer
195 views

"Converse" of Borel-Cantelli Lemma

I've stumbled upon this unusual "converse" of BC Lemma. The usual one being the one stated for example here: Proof of the converse Borel-Cantelli lemma The unusual one being: If $P(\lim \sup ...
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0 votes
1 answer
61 views

A Borel-Cantelli question

I want to apply the Borel-Cantelli lemma for say events $E_i$ which depend on some parameter $T>0$. I want a result that says something like: with probability one, only a finite number of $E_i$ is ...
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0 answers
38 views

Why do we need this extra condition on $X_n$ for convergence almost surely?

Here is a theorem on my course: If $X_n \to^{\mathbb{P}} X$ (in probability) and $\sum\limits_{n=1}^{\infty}\mathbb{P}[|X_n-X|>\epsilon] < \infty$ for all $\epsilon > 0 $ then $X_n \to X$ ...
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2 votes
0 answers
81 views

Longest run of heads (coin tossing)

The following problem comes from an old exam (introduction to probability). I have seen that there are similar questions on the forum, but most of them seem to be a bit more advanced and none of the ...
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