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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12 views

Show that $\log{M_n}/\log{n}\to1$ a.s. where $M_n = \max\{X_k \mid 1 \leq k \leq n\}$ and the $X_n$ are iid with $\mathbb{P}(X_n \geq i) = 1/i$.

Let $(X_n)_{n \geq 1}$ be a family of independent, identically distributed integer valued random variables with $\mathbb{P}(X_n \geq i) = 1/i$ for each $i \geq 1$. For each $n$, define $M_n = \max\{...
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22 views

Why sets of nonzero measure contain almost all the point from Borel-cantelli Leems?

The question referred from this discussion Borel-Cantelli Lemma "Corollary" in Royden and Fitzpatrick I understand the proof of the Leems.But i still don't get "Then almost all x ∈ R belong ...
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40 views

Sum of independent random variables is a martingale which converges almost surely

Let $ \{X_n\}_{n \ge 1} $ be a sequence of independent random variables satisfying $$ \mathbb{P}(X_n = -n^2) = 1 - \mathbb{P}(X_n = \frac{n^2}{n^2 - 1}) = \frac{1}{n^2}. $$ The question is ...
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53 views

$X_1,X_2, \ldots$ be i.i.d. Show that $\mathbb{E}|X_1| < \infty $ iff $ \frac{X_n}{n} \to 0$ a.s

Suppose $X_1,X_2, \ldots$ be i.i.d. Show that $\mathbb{E}|X_1| < \infty \Leftrightarrow \frac{X_n}{n} \to 0$ a.s I tried using Markov but I don't know anything about the $ \mathbb{E}X $. I was ...
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36 views

I don't get it why the Second Borel Cantelli Lemma here will get me to this conclusion, I don't see the link

$Let\:\left(d_1\left(w\right),...\right)\:be\:a\:sequence\:for\:which\:d_{n\:}\left(w\right)=1\:obverse\:and\:d_n\left(w\right)=0\:reverse,\:let\:A_n=\left\{d_n\left(w\right)=1\right\},\:P\left(A_n\...
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27 views

$E[|X|^r]=+\infty$ $\implies$ $P(\left\{\limsup_nY_n=+\infty \right\}\cup\left\{\liminf_nY_n=-\infty \right\})=1?!$

Let $(X_n)_n$ be a sequence of independent and identically distributed random variable, such that there exists $r>0$ such that $E[|X_1|^r]=+\infty.$ Let $(x_n)_n$ be a sequence of real numbers, $...
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32 views

Prove SLLN doesn't hold for a given series of random variables

The random variable series $\{X_k\}$ is as follows: $P(X_k=\pm k)=\frac{1}{2}k^{-\frac{1}{2}}, P(X_k=0)=1-k^{-\frac{1}{2}}$. The variables are independent. I need to prove that the Strong Law of Large ...
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63 views

Converse of Borel-Cantelli

My Prob training is rusty now.... Borel Cantelli states for part 1: If $$ \sum_{n=1}^\infty P(E_n)<\infty$$ Then $$P(E_n\text{ occurs infinitely often}) = 0$$ Is the reverse true? i.e. if: $$P(...
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42 views

Question on a application of Borel-Cantelli Lemma and answer veryfication

Let $\{A_n\}_{n\geq1}$ be a sequence events in $(\Omega, \mathcal{F}, P)$ such that $$\sum_{n=1}^{\infty} P(A_{n}\cap A^c_{n+1}) < \infty$$ and $\lim_{n\to\infty} P(A_n) = 0.$ Show that $$ P(\...
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16 views

Proving convergence a.s. of gains from a biased game

Let's say you play a game with a probability of winning $p \in \left(\frac13,\frac12\right)$ and the probability of losing $1-p$. Your initial stake is $c>0$, every time you win it doubles, every ...
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1answer
25 views

How is the Borel-Cantelli lemma applied here?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $X:\Omega\to[-\infty,\...
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39 views

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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98 views

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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1answer
43 views

For independent random variables $X_n$, how to prove $X_n / n^p \rightarrow 0$ a.s. iff $E\left(\left|X_{n}\right|^{1 / p}\right)<\infty$ for $p>0$

I am working on the following problem. Let $X_{n}, n=1,2, \ldots,$ be independent random variables with identical distribution. Show that $X_{n} / n^{p} \rightarrow 0$ a.s. if and only if $E\left(\...
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27 views

Convergence of Maximum of $n$ i.i.d. random variables in probability and almost surely

Problem: Let $\{X_k\}_{k\in\mathbb N}$ be a sequence of i.i.d. $\text{Unif}(0,1)$ random variables. Let $M_n=\max\{X_1,\dots,X_n\}$ be the maximum of the first $n$ random numbers. $\textbf{a)}$ Show ...
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41 views

Borell-Cantelli lemma

If $\sum_{n} P(|X_{n}|>n)<\infty$, then prove that the $\limsup_{n}$ $|X_{n}|/n \leq 1$ a.s. My approach Let $E_{n}=|X_{n}>n|.$ $\sum_{n}P(E_{n})<\infty$ implies $P(E_{n} \text{ i.o})=...
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53 views

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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1answer
26 views

Relation between two versions of the Second Borel Cantelli lemma

In Durett's book, the second Borel Cantelli lemma is as follows: Let $\{F_n\}$ be a filtration, and $A_n\in F_n$ be a sequence of events. Then, $\{A_n \text{ i.o.}\}=\{\omega:\sum_{n=1}^\infty P(A_n|...
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1answer
28 views

Counter example to Borell Cantelli when $\sum \mu(A_n)^2 <\infty$

I am looking for a counter example to the Borell Cantelli Lemma when $$\sum \mu(A_n)^2 <\infty$$ There is a probability solution here:A variation of Borel-Cantelli Lemma 2 but I am looking for one ...
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1answer
34 views

Showing $X_n \rightarrow 0$ a.s.

If $X_n$ is an infinite sequence of r.v.s all definied on the same space such that $P(X_n=1)=p_n$ and $P(X_n=0)=1-p_n$. How do you prove that $X_n \rightarrow 0$ a.s. if an only if $\sum_{n\geq1}p_n&...
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84 views

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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1answer
48 views

$ \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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1answer
62 views

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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39 views

Determine or find an upper bound for the limsup of a function with random variables

Determine or find an upper bound for $$\limsup_{n \to \infty} \left|\frac{\frac{2}{n} (l(\beta) - l(\hat{\beta})) - (\beta - \hat{\beta})^T \nabla^2 l(\hat{\beta})(\beta - \hat{\beta})}{(\beta - \hat{...
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1answer
31 views

For what values of $\alpha$ does $X_n$ converge almost surely to $0$?

I am working on this question: $P(X_n=n^{\alpha})=\frac{1}{n}$, $P(X_n=0)=1-\frac{1}{n}$, for what values of $\alpha$ such that $X_n$ converges almost surely to $0$? Here is what I think: $X_n$ ...
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1answer
60 views

Borel Cantelli liminf of independent random variables

Let $\{X_n\}_{n=1}^\infty$ is a succession of indepedent random variables, such that for all $n\geq 1$, $\mathbb E(X_n) =0$ and $\mathbb E(|X_n|) = 1$, Prove or disprove that $\mathbb P(\lim \inf_{n}...
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1answer
35 views

Finding the probability of a tail event

Let $(X_n)_{n \geq 1}$ independent for which we have $X_n \sim \exp(\lambda_n)$ with $\lambda_n:=\log(n+1)$. For $p>0$ consider the event $E_\rho=\left\{X_n\geq\rho \text{ for infinitely many } n\...
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48 views

Showing that $Y := |\sum_{n\geq1}X_n|$ converges with Borel-Cantelli lemmas

Let $(X_n)_{n\geq1}$ a sequence of i.i.d rv's with distribution $P(X_n=\frac{1}{n})=P(X_n=-\frac{1} {n})=\frac{1}{2}$ We need to show that $Y := |\sum_{n\geq1}X_n|$ converges a.s. It is also noted ...
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48 views

Proving that $\sum_nX_n<+\infty$ a.s. where $(X_n)_{n\geq 1}$ a sequence of independent r.v.'s

As in the titel we want to proof that $\sum_nX_n<+\infty$ a.s. where $(X_n)_{n\geq 1}$ a sequence of independent r.v.'s all on the same probability space $(E,\mathcal{E},P)$. We aslo know $P(X_n=\...
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1answer
30 views

Proving statement in probability about two sequences of random events and their infinite occurrences

This statement in probability seems definitely true, but I cannot work out how to prove it. Let $X, Y$ be measure spaces, $U \subset X$, $V \subset Y$ open sets, and $f: X \times Y \to X$ a ...
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1answer
31 views

Almost sure convergence of a series

I'm given a set of random variables $Z_n$ that have the following properties: $\mathbb{P}\left [ Z_n = \frac{1}{n} \right ] = \frac{1}{n^2} $ and $\mathbb{P}\left [ Z_n = 0 \right ] =1- \frac{1}{...
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2answers
48 views

Proving that a series of iid random variables diverges

so I am working on a problem and it basically boils down to showing the following: Let $Y_1, Y_2,...$ be i.i.d. random variables on $(\Omega, \mathcal{F}, \mathbb{P})$ satisfying $\mathbb{P}(Y_1=-1)=...
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1answer
27 views

Almost sure convergence of order random variables

OK, let $X_{1},...X_{n}$ be random variable independts and distributed in the same way. Let $m=inf X_{1}$ and $M=supX_{1}$, that means $\forall a>m, {\cal P}(X_{1}<=a)>0$ and $\forall b<M, ...
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47 views

Existence of subsequence $\{n_k\}$ tending to infinity such that $P(\cap_k A_{n_k}>0)$

I started this year to study probability and I'm using A Probability Path of Resnick, I'm trying to solve this exercise but I can't see the link between the Borel-Cantelli's Lemma that I studied and ...
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1answer
31 views

Proof of Reverse Borel Cantelli Lemma

I read a proof on The converse Borel Cantelli lemma unfortunately I can't upload a photo from my app I don't know why In the proof they use $P(lim inf {A_n}^c) = {lim}_{n \rightarrow \infty} P(\...
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1answer
45 views

Probablility that the sum occurs infinitely often

I encountered a problem recently, which was stated as follows: "suppose $x_i$, $x_2$, ... are a sequence of independent random variables, and let $S_n = x_1 + ... + x_n$. What is the probability that $...
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46 views

A variation of Borel-Cantelli Lemma 2

This problem is from my midterm. I have not solved it since. "Given a measure space $(X,M,\mu)$ and measurable sets $A_1,A_2,\dots$ s.t. $$\sum \mu(A_i)^2<\infty. $$ Give an example where the ...
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1answer
103 views

Kolmogorov’s Three-Series Theorem

Consider the sequence r.v. $X_i's, i \geq 2$, $$P(X_i = i^2) = P(X_i = -i^2) = 1/i^2 \ \text{and} \ P(X_i = (-1)^i) = 1- 2/i^2.$$ Consider $S_n = \sum_{i=2}^{n}X_i$. What is the almost sure limit of $...
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1answer
57 views

Prove this exercise from K.L. Chung using just Borel Cantelli Lemma

Prove that probability of convergence of sequence of independent r.v. is either $0$ or $1$. Proving the convergence is $0$ is straightforward, using B-C lemma. But for proving it to be $1$, we need ...
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41 views

Borel-Cantelli problem: Prove sum of probabilities is finite

I am looking for good solutions to this problem. Could you please help me with this? Any solution would be appreciated. Let $\mathit{X_n}$ be independent random variables. Show that if $\mathit{X_n}...
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1answer
47 views

Show that if $\mathbb{P}(X_{k}\neq 0\ \text{i.o.})=0$, then $\sum_{k=1}^{n}X_{k}$ converges almost surely.

By i.o., I mean the infinity often. As asked in the title, I am working on the statement below: Let $X_{k}$ be mutually independent random variable and set $S_{n}:=\sum_{k=1}^{n}X_{k}$, show that ...
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1answer
32 views

probability that joint events are infinitely often?

If set sequences $A_1, A_2, \cdots$, $B_1, B_2, \cdots$ consist of measurable sets in the sigma-algebra $Q$. Suppose $P(A_k \text{ infinitely often }) = 1$, and $P(B_k^c \text{ infinitely often }) = 0$...
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1answer
28 views

Kolmogorov three series for sequence of exponential

If $X_1, X_2,..$ is a sequence of independent random variables, each exponential such that $\mathbb{E}[X_n] = \frac{1}{\lambda_n}$. 1) If $\displaystyle \sum_{n=1}^\infty \frac{1}{\lambda_n} < \...
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2answers
27 views

Showing a probability space

I have recently been exposed to probability measure theory and found an exercise in my textbook which I am not able to solve. I have to show that ($\mathbb R, \mathscr B(\mathbb R), \mu_X $) is a ...
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1answer
252 views

Using the first and second Borel-Cantelli Lemma to find necessary and sufficient condition for convergence in probability ($98\%$ solved)

I am working an exercise with five parts, and I've solved most of them, but still have some little but nontrivial confusions. The parts (b)-(e) coincides with Durrett 1.6.15 or Durrett 2.3.15, and ...
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1answer
91 views

$P(\limsup A_n)=1 \Leftrightarrow \sum_{n=1}^{\infty} P(A \cap A_n) = \infty\; \forall A, P(A)>0$

Let $\{A_n\}$ be a sequence of independent events. How to prove that $$P(\limsup A_n)=1 \Leftrightarrow \sum_{n=1}^{\infty} P(A \cap A_n) = \infty\; \forall A, P(A)>0?$$ As the $A_n$ are ...
3
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1answer
104 views

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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1answer
60 views

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 ...
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1answer
39 views

Borel Cantelli Problem - Normal Random Variables

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>...
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1answer
25 views

Borel-Cantelli Lemma in Terms of Transformation

I am studying dynamical system, and by accident I found a paper here https://arxiv.org/pdf/math/9912178.pdf In the introduction part, the authors firstly state the Borel-Cantelli lemma, then they ...

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