Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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).

0
votes
1answer
28 views

Does $P(\liminf_{n \to \infty}\{|X_{n}|\leq \epsilon\})=1\iff \exists N \in \mathbb N, |X_{n}|\leq \epsilon, \forall n \geq N, P-$a.s.

Background to my question: Given that $(X_{n})_{n}$ are random variables on $(\Omega, \mathcal{F}, P)$ and for $\epsilon > 0$: $\sum_{n \in \mathbb N}P(|X_{n}|>\epsilon)<\infty$ It follows ...
2
votes
1answer
26 views

$\limsup_{n \to \infty}\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\}\subseteq \{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\leq\epsilon\}$

I recently saw an assertion made that $$\bigcap_{m \in \mathbb N} \bigcup_{n \geq m}\left\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\right\} \subseteq\left\{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\...
0
votes
0answers
35 views

Question on $\liminf A_{n}$ and $\limsup A_{n}$

Say I have the following event $\{\limsup_{n \to \infty} |X_{n}|> \epsilon\}$, and that $P(\{\limsup_{n \to \infty} |X_{n}|> \epsilon\})$, I also realize that $\{\limsup_{n \to \infty} |X_{n}|&...
1
vote
1answer
27 views

Show that $n^{\alpha}X_{n} \xrightarrow{n \to \infty} +\infty$ almost surely

Let $(X_{n})_{n}$ be a sequence of random variables that are identically distributed on $\mathcal{U}(0,1)$. Furthermore, let $\alpha > 1$. Show that $n^{\alpha}X_{n} \xrightarrow{n \to \infty} +\...
1
vote
1answer
23 views

Showing $\limsup_{n \to \infty}\frac{X_{n}}{\log(n)}=1$ a.s.

Let $(X_{n})_{n}$ be independent, identical random variables such that $X_{n}$~$\exp(1)$ Show that $\limsup_{n \to \infty}\frac{X_{n}}{\log(n)}=1$ a.s. I am given a hint to look at $P(\frac{X_{n}}{\...
1
vote
1answer
31 views

Have to use Borel-Cantelli appropriately here

An urn contains one black and one white ball. Someone draws a ball and then places it back and adds as many white balls as are currently in the urn. This is process is repeated infinitely often. ...
2
votes
1answer
37 views

Question on the application of the Borel-Cantelli lemma

I have a question on the first proof on this pdf. Essentially, it is trying to prove the following statement: Let ${f_n}$ be a sequence of measurable functions on $[0, 1]$ with $|f_n(x)| < ∞$ for ...
1
vote
1answer
19 views

Prove that the infinite sum of folded normal variables diverges almost surely

Let $\xi_n \sim N(0,1)$ and $A_n = \frac{\sqrt 2}{\pi n} \xi _n$. Show that $\sum_{i=1}^\infty |A_i| = \infty$ $\mathbb P$-a.s. MY ATTEMPT: Usually when trying to prove or refute almost sure ...
1
vote
1answer
55 views

Find $\lim\limits_{n \to \infty}\mathbb P\left(\frac1{\sqrt n}\sum\limits_{i=1}^nX_i\le x\right)$ if $P(X_n=0)=1-\frac1{n^2}$

Let $(X_{n})_{n}$ be independent random variables such that $\mathbb P(X_{n}=n)=\mathbb P(X_{n}=-n)=\frac{1}{2n^2}$ and $P(X_{n}=0)=1-\frac{1}{n^2}$. Find $\lim_{n \to \infty}\mathbb P(\frac{1}{\...
0
votes
2answers
29 views

$\sum_{n=1}^{\infty}p_{n}<\infty \iff X_{n} \to 0$ a.s.

Let $(p_{n})_{n}\subset [0,1]$ and $(X_{n})_{n}$ independent random variables, so that $X_{n}$~ $Ber(p_{n})$ Prove that: $\sum_{n=1}^{\infty}p_{n}<\infty \iff X_{n} \to 0$ a.s Ideas: "$\...
0
votes
1answer
31 views

Almost sure convergence and Borell - Cantelli Lemma 2

Suppose we have the following random variable: $X_n = n$ with probability $\frac{1}{n}$ and $0$ with probability $1-\frac{1}{n}$. We we can define this variable on the probability space $([0,1], \...
1
vote
1answer
55 views

Show that $n \mathbb{P}\{|X_1| \geq \epsilon \sqrt{n}\} \to 0$

Let $(X_n)_n$ be a sequence of identically distributed random variables with $\mathbb{E}X_1^2 < \infty$. Show that $$\lim_{n \to \infty} n\mathbb{P}\{|X_1| \geq \epsilon \sqrt{n}\} = 0$$ for all ...
2
votes
3answers
55 views

A.S. convergence of sum of square-integrable independent random variables with summable variation

I'm working on the following exercise from Achim Klenke's "Probability Theory: A Comprehensive Course" (exercise 6.1.4): Let $X_1, X_2, \ldots$ be independent, square integrable, centered random ...
1
vote
1answer
38 views

Equivalence of the condition that the supremum of i.i.d. RVs are finite a.s.

I am proving the following : Suppose $\{X_n : n\in\mathbb{N}\}$ are i.i.d. random variables. Then $P(\sup_{n\in\mathbb{N}}X_n < \infty) = 1$ if and only if $ \sum_{n\in\mathbb{N}}{P(X_n > M)} &...
4
votes
1answer
33 views

Non convergence of a series of random variables

Question: Let $(X_n), n\in\mathbb{N}$ be a sequence of independent r.v.s such that $P(X_n=n^4)=\frac{1}{n^4}$ and $P(X_n=-1)=1-\frac{1}{n^4}$. Study the a.s. convergence of $S_n=\sum_{i=1}^n X_n$ ...
1
vote
0answers
45 views

First Borel-Cantelli-Lemma - Two different proofs

The Borel Cantelli Lemma states that for a sequence of events $(A_n)$ with finite sum $\sum_{k=1}^{\infty}P(A_k)< \infty$ the probability of infinite many events happening is $0$, thus $P(\lim\sup_{...
0
votes
1answer
37 views

Inequality in proof of 2nd Borel-Cantelli Lemma

At some point in the proof of the second Borel-Cantelli Lemma the the following inequality is mentioned: $$...=\exp\bigl ( \sum_{m=n}^k\log(1-P(A_m)\bigr ) \leq \exp \bigl (-\sum_{m=n}^kP(A_m) \bigr)$...
1
vote
1answer
67 views

IID random variables $(X_n)$ have $\sum e^{X_n} c^n < \infty$ a.s.

I'm working on the following exercise: Let $X_1, X_2, \ldots$ be i.i.d. nonnegative random variables. By virtue of the Borel-Cantelli lemma, show that for every $c \in (0,1)$, $$ \sum_{n=1}^\...
0
votes
1answer
24 views

For random variables $(X_k)_{k=1}^\infty$ in $\mathbb{R}$, find $(c_k)_{k=1}^\infty$ such that $P(\lim (X_k/c_k)=0)$

I'm stuck with the following problem: Let $(X_k)_{k=1}^\infty$ be a sequence of real-valued random variables. Show that there is a real sequence $(c_k)_{k=1}^\infty$ such that $$P\left(\lim_{k\...
1
vote
1answer
33 views

Almost sure convergence of $|X_n|/n$ for a sequence of i.i.d r.v.'s

Let $\{X_n: n \ge 1\}$ be a sequence of i.i.d random variables with $\mathbb{E}[|X_1|] < \infty$, and $\mathbb{E}[X_1] \neq 0$. Show that $$\frac{|X_n|}{n} \to 0 \quad \text{almost surely.}$$ ...
0
votes
1answer
32 views

Prove that $M_n/\log n\to 1$ a.s. where $X_i$ are a sequence of i.i.d $\text{exp}(1)$ random variables and $M_n=\max_1^n X_i$

Question Let $(X_n)_{n\geq 1}$ be an i.i.d sequence of random variables with $P(X_1>x)=e^{-x}$ and put $M_n=\max_1^n X_i$. Then $M_n/\log n\to 1$ a.s. My attempt I have shown that $\limsup X_n/...
1
vote
1answer
72 views

If a sequence of independent random variables converges almost surely to a random variable, then that limit is almost surely a constant

Let $\{X_n\}$ be a sequence of independent random variables converging almost surely to a random variable $X$. Then how to show that $X$ is almost surely a constant ? I think I somehow have to apply ...
2
votes
1answer
31 views

How to show convergence a.s. when sum of $P(A_n)$ is $\infty$ and the sequence is not independent

There is a sequence of random variables defined by $$Y_n = \Big(\Big|{1-\frac\Theta \pi}\Big|\Big)^n$$ where $\Theta\sim\mathrm{unif}[0,2\pi].$ I have shown that the sequence converges to $0$ in ...
1
vote
1answer
48 views

Borel Cantelli Lemma for non independent random variables

Suppose $Y_n$ is a sequence of random variables, not independent, which converges to $0$ in probability. Define a set $A_n = \{|Y_n|>\epsilon\}$. If $\sum_1^\infty P(A_n) = \infty$, can it be ...
0
votes
0answers
9 views

Wiener process infinitely many times in some range

I have wondered about the probability that Wiener process $W_t$ will be infinitely many times in some interval, let's say $[-1,1]$. Is it possible to obtain? Of course it could be $0$ or $1$, but how ...
4
votes
1answer
76 views

Prove that $\liminf \frac{M_n}{\sqrt{2\log n}}\geq 1$, where $M_n=\max_{1}^n X_i$ and $(X_i)$ is a sequence of i.i.d standard normal random variables

Question Let $(X_n)_{n\geq 1}$ be an i.i.d sequence of standard normals. Show that with probability one $\liminf \frac{M_n}{\sqrt{2\log n}}\geq 1$, where $M_n=\max_{1}^n X_i$. My attempt Given $\...
3
votes
1answer
95 views

Borel-Cantelli and “infinitely often”

The problem: Let $(X_n)_{n\geq 1}$ be a real-valued sequence of i.i.d. random variables and let $c > 0$. Use Borel-Cantelli's lemma to show that $$\sum_{n=1}^\infty P(X_n^2 > n) < \infty \...
2
votes
0answers
31 views

probability of $\left[\log_{2}n \right]$ consecutive tails

We consider the experiment on tossing a fair coin infinitely many times, and let $T_{n}$ be the even that the $n^{\text{th}}$ coin is a tail. The problem is to determine (here $\left[ x \right]$ ...
0
votes
0answers
27 views

Define two sequences of i.i.d random variables

Could you kindly explain in very simple words, or through examples, what these two sequences mean? $A_n :=\{X_n > X_i, \;for\; i∈\{0,...,n−1\}\},\\ B_n := \{A_n ∩ A_{n+1}\}.$ I just know that $(...
3
votes
1answer
54 views

Extension of Borel-Cantelli in Probability Theory

I am working on a problem regarding an extension of the Borel-Cantelli lemma that goes as follows: Let $E_1, E_2, ...$ be an arbitrary sequence of sets. It is known that $\lim_{n \to \infty}P(E_n) = ...
1
vote
1answer
113 views

Application of Borel-Cantelli Lemmas

Can you help me solving the following exercise? I should use the Borel-Cantelli Lemmas, but I don't know how. Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and let $(X_n)_{n \in \...
1
vote
0answers
39 views

Show that $\mathbf{P}(\cup_{n\geq 1}A_n)=1$ iff $\mathbf{P}(A_n\text{ i.o.})=1$.

This is a typical problem in probability theory, including Ph.D Qualifying Exam. Suppose $\{A_n,n\in \mathbb{N}\}$ is an infinite family if independent events with $\mathbf{P}(A_n)<1$. Show that $\...
1
vote
0answers
42 views

Is a sequence diverging almost surely to infinity almost surely positive?

I have proved that a sequence of random variables $(M_n)_{n\in\mathbb N}$ diverges to $+\infty$ almost surely. I.e I have proved that $$\bigcap_{c\in\mathbb Q^+}\bigcup_{N=1}^{\infty}\bigcap_{n=N}^\...
0
votes
0answers
52 views

Problem from Feller

The problem from Feller's Probability vol.1, chap. VIII that I'm trying to prove without progress: From the law of iterated logarithm conclude: With probability one it will happen for infinitely many ...
2
votes
1answer
59 views

Rate of convergence of sum of random variables

I'm working on the following problem and I'm a little stuck Suppose $X_1,X_2,\dots$ are iid. (a) If $E|X_1|^\alpha$ is finite for some $\alpha>0$, show that $\max_{1\le k\le n} |X_k|/n^{1/...
1
vote
1answer
95 views

Show $\liminf\limits_{n\to \infty} X_n = 0$ almost surely for $X_n$ uniform on $(0,n)$ and $(X_n)$ independent

Let $(X_n)_{n \in \mathbb N}$ a sequence of independent random variables, where $X_n$ is uniformly continuous distributed across $(0,n)$. Show that $\liminf_{n\to \infty} X_n = 0$ almost surely. Hint: ...
2
votes
1answer
153 views

Question regarding Borel-Cantelli lemma

Let $X_1,...X_n$ be a sequence of random variables such that $X_n=1 $ or $0$ and $P(X_1=1) \geq \alpha$ and $P(X_n=1|X_1,...X_{n-1}) \geq \alpha$ for $n=2,3,...$ where $\alpha >0$ I need to show ...
5
votes
2answers
67 views

Application of Borel Cantelli Lemma 1

$\textbf{Problem} $ Let $m$ be the Lebesgue measure on $\mathbb{R}$. Let $f_n: \mathbb{R} \rightarrow [0,\infty)$ be sequence of Lebesgue measurable functions. Show that there is a sequence $c_n$ of ...
-1
votes
1answer
107 views

Kolmogorov's Strong Law and Almost Sure Convergence

I would like a help in the following problem $(X)_{n \geq 1}$ iid, $E(X_i) = \xi$ and $Var(X_i) = \sigma_i$. Show that $\dfrac{X_1+X_2+\cdots+X_n}{n} \rightarrow \xi$ with probability 1, ...
0
votes
1answer
26 views

Convergence in probability and almost-sure convergence

I have the following sequence of independent r.v. $$\mathbb{P}(\{X=n\})=\big(\frac{1}{n^2}\big), \mathbb{P}(\{X=0\})=\big(1-\frac{1}{n^2}\big)$$ and also, $S_n = \sum_{i=1}^{n} X_i$. I am denoting ...
2
votes
1answer
115 views

Proof of the converse Borel-Cantelli lemma

BOREL CANTELLI LEMMA : For arbitrary sequence of events $\{A_n\}$, we have $$\sum_{n=1}^{\infty} P(A_n)<\infty \implies P(\limsup A_n)=0$$ CONVERSE BOREL CANTELLI LEMMA : For independent sequence ...
0
votes
0answers
113 views

relationship between infinitely often and almost sure convergence

To find a relationship between $P($A_n $ i .o)$ and almost convergence we generally use the Borel Cantelli lemma. But let say $A_n$ = ($X_n \neq $ $Y_n$) where $Y_i= X_i I(|X_i|\leq i)$. It is ...
3
votes
1answer
157 views

Almost sure convergence of sequence of discrete uniforms to continuous uniform

Let $X_n$ $(n=1,2,\dots)$ be a sequence of discrete random variables, where the distribution of $X_n$ is the discrete uniform over $\{0, 1/n, 2/n,\dots,1 \}$. Let $U$ be a random variable whose ...
1
vote
1answer
40 views

Tight asymptotic upper-bound via Chebyshev

I have a sequence $X_n$ of independent r.v. such that $X_i \sim \mathcal{N}\left(0,\frac1i\right)$. Then I have that $Y_i = e^{X_i}$ and $Z_n= \prod_{j=1}^n Y_i$. I need to give a tight, asymptotic ...
0
votes
1answer
94 views

Use Borel-Cantelli to determine $\operatorname{lim sup_{n \to \infty}}$ for some i.i.d. $X_n$

Let $X_1, X_2, \ldots$ be i.i.d. nonnegative random variables. I want to use the Borel–Cantelli lemma, to show that \begin{equation*} \limsup_{n\rightarrow\infty} \frac{1}{n} X_n = \begin{cases} ...
4
votes
0answers
32 views

Coin flipping experiment [duplicate]

This was an exercise in an old exam in probability theory without a solution. We are flipping a coin repeatedly. The probability of tail is $p$ < $\frac{1}{2}$. Let $A_{k}$ for $k \geq 2$ be the ...
2
votes
1answer
93 views

constructing a zero-mean martingale $\to -\infty$ a.s. [closed]

I'm reading a book over martingale theory and there is a exercise where I'm interested in a solution. Let $X_{1},X_{2},...$ be independent random variables with \begin{align} \mathbb{P}(X_{n} = -1) &...
1
vote
0answers
26 views

Submartingale bounded difference [duplicate]

Let $X_n$ be s submartingale and $\zeta_n = X_n - X_{n-1}$ with $\text{sup}\ X_n < \infty$ and sup $\mathbb{E}[\zeta_n^+] < \infty$. Show $X_n \rightarrow X$ a.s. For some $\epsilon > 0$, ...
0
votes
0answers
58 views

Reverse Borel Cantelli

My question is regarding the Borel Cantelli Lemmas, what can we say about their reverse statements? $X_{i}$ independent in what follows. For example say $P(X_{n}>n$ i.o$)=0 $ can we say $\sum P(...
2
votes
1answer
57 views

Proving almost sure convergence (help understanding a step in a proof)

We have $$|V_{n+1}-V^\prime_{n+1}|=\prod_{m=1}^n|W_m||V_1-V^\prime_1| \tag{1}\label{1}$$ where $P(|W|=1)\ne 1$ (with $W\in[-1,1]$) and $W_n$s are iid and independent of $V$s and $V^\prime$s. Show ...