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).

Filter by
Sorted by
Tagged with
1
vote
0answers
47 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 ...
0
votes
1answer
37 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 ...
0
votes
0answers
31 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 $X_n \to X$ ...
2
votes
0answers
54 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 ...
4
votes
1answer
51 views

How to apply Borel-Cantelli Lemma to find if something converges almost surely to 0.

Suppose that $Z_1,Z_2,...$ are random variables with $Z_n∼Exp(1)$. (We do not assume that these random variables are independent.) Show that $Z_n/\big(\ln^2(n)\big)$ converges to $0$ almost surely. ...
2
votes
0answers
91 views

Borel-Cantelli Application to show a set has measure $0$ my attempt was incorrect

Let $\alpha > 2$ be a real number. Define $$E = \{x \in [0,1] \ s.t. \ |x - \frac{p}{q}| < \frac{1}{q^{\alpha}} \ \forall p,q\}$$ Prove $m(E) = 0$ Hint: fix $p,q$ and calculate the measure of ...
1
vote
1answer
28 views

Borel Cantelli, Bernoulli trials

Suppose we conduct a countably infinite number of independent Bernoulli trials, such that each trial results in $0$ w.p. $p$, or $1$ w.p. $q\equiv1-p$. What is the probability that the sequence $(1,0,...
1
vote
0answers
28 views

Does $X_n \ \xrightarrow{a.s.}\ X$ not necessarily imply that $\sum_{n=1}^{\infty} P(|X_n-X| > \epsilon) < \infty$? Is the below an example?

Question taken from https://www.probabilitycourse.com/chapter7/7_2_7_almost_sure_convergence.php Consider the sample space $S=[0,1]$ with a probability measure that is uniform on this space, i.e., $\...
1
vote
1answer
31 views

Is $\{A_{n} \text{ finitely often} \}$ the complement of $\{A_{n} \text{ infinitely often} \}$

I tried to think about this using De-Morgan's laws: $\{A_{n} \text{ i.o} \} = \bigcap_{m\ge 1} \bigcup_{n\ge m} A_n$ so $\{A_{n} \text{ i.o} \}^c = \bigcup_{m\ge 1} \bigcap_{n\ge m} {A_n}^c$ once you ...
3
votes
1answer
44 views

What do these alternative formulations of the second Borel-Cantelli lemma (Durrett theorem 4.3.4 and 4.5.5) say?

I am reading Durrett and I don't understand what do these formulations of the second Borel-Cantelli lemma say: Specifically, what does $P(B_n|F_{n-1})$ mean?
0
votes
1answer
45 views

Borel-Cantelli lemma, exponential distribution problem

Good evening, I am currently solving an exercise : Let Xn be a sequence of independent random variables, each with the exponential distribution with rate $\frac{1}{\lambda}$. I want to prove the part :...
0
votes
2answers
44 views

Convergence of series $\frac{x^{n}}{\sqrt{n}}$ when $0<x<1$

I was stuying 1-D random walk example in Borel-Cantelli Lemma Event $A_n$ = We return back to origin in 2n steps Probability of taking right step at each point = p Probability of taking left step at ...
2
votes
1answer
66 views

Converge almost surely with sums of Bernoulli random variable

Let $A_{1},A_{2},...$ be a sequence of independent events, and let $b_{n}=\sum_{i=1}^{n}P(A_{i})$. If $b_{n}\to \infty$, then $\frac{1}{b_{n}}\sum_{i=1}^{n}I_{A_{i}}\to 1$ a.s. How can I prove it? ...
1
vote
0answers
23 views

Proof of Borel - Cantelli by contradiction.

I tried proving the following lemma: If $(A_n)_n$ is a sequence of events in a probability space s.t $\sum_{n=1}^{\infty}\mathbb{P}[A_n]<\infty$, then $\mathbb{P}[\text{limsup}A_n]=0$. My attempt: ...
1
vote
1answer
42 views

Almost sure convergence on integral of Poisson process

Let $N_t$ be a Poisson process with intensity $\lambda$. Then how to show that $$ \lim_{t\rightarrow 1} (1-t) \int_0^t \frac{2}{(1-s)^2}dN_s = 0 $$ almost surely? What I have done. Since for some $\...
3
votes
1answer
41 views

Sufficient Conditions for Borel-Cantelli

The first Borel-Cantelli Lemma states If $\sum \mathbf{P}(A_n) < \infty$ then $\mathbf{P}(A_n i.o.) = 0$ Question : If $\mathbf{P}(A_n) \rightarrow 0$ and $\sum \mathbf{P}(A_{n+1} / A_{n}) < \...
1
vote
1answer
44 views

How is Borel Cantelli Lemma being applied in this case?

I am taking a course in Probability Theory and in a proof, the professor made the following claim: If for all $\epsilon>0$, the sum $\sum_{n\geq 1}P(|X_n-E[X_n]|> n\epsilon)<\infty$ then by ...
2
votes
0answers
46 views

Calculating $P(\sum X_i < \infty)$ for $X_i\sim U(0,1)$

$\sum X_i < \infty$ is a tail event, $X_i$'s are independent, so by the Kolmogorov $0-1$ theorem, $P(\sum X_i < \infty)$ can be either $0$ or $1$. But which one is it? I tend to think it's $0$. ...
3
votes
0answers
33 views

When does the sequence $1_{A_n} - 1_{A_n ^C}$ weakly converge to $0?$

Let $(\Omega,f,p)$ be an abstract probability space. Consider the variables $$ X_n = 1_{A_n} - 1_{A_n^C} \in L^p(\Omega)$$ I am interested in charactarizing when $X_n \to 0$ weakly, meaning that $E[Y ...
2
votes
1answer
44 views

Convergence almost surely question

Suppose you have a sequence of independent random variables {$X_i, i\geq1$}, such that $$\Bbb P(X_i=i^2 -1)=i^{-2},$$ $$ \Bbb P(X_i=-1)=1-i^{-2}.$$ Then, $\frac1n \sum_{i=1}^n X_i$ converges almost ...
0
votes
1answer
25 views

$X$ RV $X(S)=s$. Prove $X_n\underset{a.s}{\to}X$.

Let $([0,1],\mathcal{B}_{[0,1]},\lambda)$ probabilty space and let $X_n(s)=\frac{n}{n+1}s+(1-s)^n$ a sequence for RVs for every $s\in[0,1]$. and let $X$ RV $X(S)=s$. Prove $X_n\underset{a.s}{\to}X$. ...
0
votes
1answer
29 views

>Let $X_1,\ldots,X_n$ indepent variable RV $X_n\sim Bern(1/n)$ Does the $X_n\underset{a.s}{\to}0$

Let $X_1,\ldots,X_n$ indepent variable RV $X_n\sim Bern(1/n)$ Does the $X_n\underset{a.s}{\to}0$ I tried to use Borel-Cantelli but I get that $\sum_{i=1}^{\infty}\mathbb{P}(A_n^{\epsilon})=\infty$ ...
1
vote
1answer
42 views

Clarification on convergence almost surely

The three aspects within convergence in distributions include: Convergence in distribution Convergence in probability Convergence almost surely Convergence in distribution and in probability ...
0
votes
1answer
27 views

The divergence of a series implies the divergence of limit superior of absolute value of sums of random variables

I'm here for another problem. I need to show the following: If $\{X_n\}$ is a sequence of independent random variables, and $S_n=\sum_{k=1}^{n}X_k$ then $$\limsup_{n}|S_n|=\infty \quad a.s. \text{ if }...
0
votes
1answer
64 views

Why almost sure convergence holds if $X_n = n$ w.p. $1/n$?

I've encountered these two examples (used to show how a.s. convergence doesn't imply convergence in $R$th mean and visa versa). In one case we have a random variable $X_n = n$ with probability $\frac{...
2
votes
0answers
28 views

Measure of a set with digits resemblance to $\pi$

Let $I= (3,4)$, and write the decimal expansion of $x\in I$ in the form $x= 3.d_1d_2d_3 . . .$ such that $d_k$ is not eventually 9, (this makes the decimal representation of $x$ unique). Let $c_k$ be ...
2
votes
1answer
30 views

Theorem on lim sup of max of random variables.

Let $X_1,X_2,\ldots$ be iid random variables, then we are trying to prove that: $$\limsup_{n\to\infty} \frac{1}{n} \max_{k\leq n} \lvert X_k \rvert = \begin{cases}0 &\textrm{if } E\lvert X_1\...
1
vote
0answers
41 views

Understanding a definition of infinitely often

My textbook states that the definition for $A_{n}$ occurs infinitely often is: $\limsup\limits_{n \to \infty} A_{n} = \cap_{n=1}^{\infty}(\cup_{m \geq n}A_m) = \lim_{n \to \infty} (\cup_{m \geq n} A_m)...
2
votes
2answers
48 views

Sum of Indicators: $\sum_{n=1}^{\infty} \mathbb{1}\{X_1^2/\alpha^2 \geq n\} = \left(1 + \left\lfloor\frac{X_1^2}{\alpha^2}\right\rfloor\right)$

I was working on proving the following proposition: Suppose $(X_n)$ is some iid sequence of random variables such that $\mathbb{E}(X_1)^2 < \infty$. Then for every $\alpha > 0$ we have $$\...
1
vote
0answers
50 views

Borel-Cantelli Application Question

I have a question (at $\star$) regarding the solution to the question below: Question. Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of random variables. Show that there exists a sequence of numbers $\{...
1
vote
1answer
55 views

What's the probability for k consecutive 1s? I can use Borel-Cantelli

Let ${X_{i}}$ be an infinite sequence of independent random variables with $p = P(X_{i} = 1) = 1−P(X_{i} = −1)$. Let $A_{k}$ be the event that $(X_{2^k},X_{2^k+1},X_{2^k+2},...,X_{2^{k+1}−1})$ ...
1
vote
0answers
22 views

Maximum length of consecutive 1's in a random 01 string

For every positive integer $n$, let $s_n$ be a uniformly distributed random 01 string with length $n$. For a positive integer $k\leq n$, we say $s_n$ has $k$ consecutive 1's if it contains a length-$k$...
0
votes
2answers
33 views

Choosing positive reals $\alpha_i$ for $a_iX_i \sim N(0, a_i^2)$ such that $a_iX_i \stackrel{a.s.}{\to} 0$

Suppose we have a decreasing sequence $\{\alpha_i\}_{i=1}^{\infty}$ where $\alpha_i \in \mathbb{R}^{+}$. That is $$\mathbb{R}^{+} \ni \alpha_1 \geq \alpha_2 \geq \cdots > 0.$$ Further, let $\{X_i\}...
0
votes
0answers
14 views

Probability of double records

Let us consider a sequence of iid random variables $(X_n)_n$ with continuous distribution on $[0,1]$. For $n \ge 1$, we say that there is a record at time $n$ if $X_n > \max(X_0, \ldots, X_{n-1})$. ...
1
vote
1answer
21 views

Show that this sequence does not obey the SLLN

The exercise reads: Let $(\sigma_n)$ be a sequence of real numbers for which $$ \sum_{n=1}^{\infty} \frac{1}{n^2} \sigma^2_n=\infty$$ Prove that there exist an independent sequence $(X_n)$ of ...
0
votes
1answer
47 views

Show that two sequences both converge or both diverge almost surely

Let $(X_n)$ be a sequence of independent real random variables and $(\tau_n)$ be a null sequence of real numbers. Prove that either each of the sequences $(X_n)$ and $ \big (\tau_n\sum_{i=1}^n X_i \...
3
votes
2answers
126 views

An application of Borel-Cantelli Lemma?

I decided to dust off my measure theory notes and try some problems. I saw that I left this homework problem blank a few years ago. Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. ...
0
votes
1answer
95 views

$\lim_{n} P(U_n)=0,$ $\sum_{n}P(U_n \cap U_{n+q_n}^c)<+\infty \implies P(\limsup_nU_n)=0$

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(U_n)_n$ a sequence of elements in $\mathcal{F},(q_n)_n$ a sequence in $\mathbb{N}^*$ such that $\lim_{n \to+\infty} P(U_n)=0$ and $\sum_{n\in \...
1
vote
1answer
79 views

Using Borel Cantelli lemma to show that the set of convergence of non degenerate independent random variables has measure zero.

I'm trying the following: If $X_1,\dots, X_n,\dots, $ are non degenerate independent and identically distributed random variables; then \begin{equation*} \mathbb{P}\left(X_n \text{ converge }\right)=...
1
vote
1answer
53 views

Borel-Cantelli Lemma - is the measurability assumption necessary?

I've seen this version of the Borel-Cantelli Lemma in Stein and Shakarchi: Let $(E_k)_{k=1}^{\infty}$ be a countable family of measurable subsets of $\mathbb{R}^d$ satisfying $$\sum_{k=1}^{\infty}m(...
0
votes
0answers
36 views

Frequency of events when probability goes to $0$

Let $X_1,X_2,\dots$ be sequence of random Bernoulli variables. Thanks to the Borel-Cantelli lemma, if the series of expectations is finite, then almost all elements are $1$s almost surely. If the ...
2
votes
0answers
63 views

a.s limit of the maximum of a sequence of Poisson distribution

Let $(X_n)_n$ be a sequence i.i.d random variable following Poisson distribution $P(\lambda).$ Prove that, almost surely, $$\lim_n\frac{\max_{1\leq k \leq n}X_k\ln(\ln(n))}{\ln(n)}=1.$$ Hint: Use ...
1
vote
0answers
43 views

Probability of finite heads in coin flips

$\mathbb{P}[X_{n}$ is head $] = p_{n}$. $F$ is the event that only finitely many heads are observed. How do I prove $\mathbb{P}[F]=1$ iff $\sum_{n=1}^{\infty} p_{n} <\infty$ I am confused about ...
8
votes
2answers
269 views

Necessary and sufficient condition for convergence of series

I'm solving this exercise in Klenke's book: Let $X_1,X_2, \dots $ 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}^\infty ...
6
votes
2answers
156 views

Prove that if the series is convergent then the law of large numbers hold.

Let $(X_n)_{n \geq 1}$ be a sequence of pairwise independent random variables such that : $$\sum_{n=1}^{\infty} n^{-1} P\left\{\max _{1 \leq m \leq n}\left|\sum_{k=1}^{m}\left(X_{k}-E X_{k}\right)\...
1
vote
1answer
58 views

Proof of almost sure convergence of sum of iid random variables

I tried to prove the following lemma: Let $X_1,X_2,\ldots$ iid nonnegative random variables with $E[X_1]=\infty$ and let $a\in(0,1)$, thus follows $\sum\limits_{n=1}^\infty a^n exp(X_n)=\infty$ almost ...
0
votes
1answer
66 views

Counter example to Borel Cantelli Lemma when assuming only that probabilities converge to zero

As I understand it, the (First) Borel Cantelli Lemma says that if $$\sum_{n=1}^\infty P\{E_n\} < \infty$$ then $$P\left\{\bigcap_{m=1}^\infty \bigcup_{n=m}^\infty E_n\right\} = 0.$$ Why is it not ...
0
votes
1answer
44 views

Show that the set has measure zero, using Borel-Cantelli lemma.

(Tao Vol.2, P.200, Exercise 8.2.7) Let $p > 2$ and $c > 0$. Using the Borel-Cantelli lemma, show that the set $$\left\{x \in [0,1] : \left|x - \frac{a}q\right| \le \frac{c}{q^p} \quad\text{for ...
2
votes
1answer
141 views

Convergence almost surely of Brownian Motion

If $B_t \sim N(0,t)$ then, intuitively, for any fixed $\varepsilon$, as $t \to \infty$, the probability that $B_t$ will be observed within the $[-\varepsilon, \varepsilon]$ interval should converge ...
0
votes
1answer
75 views

Determine the probability of the event $\limsup A_n$ when given $P(A_n)$

Let $(X, A, P)$ be a probability space and $(A_n)_{n\in \mathbb N}$ a sequence of events in $A$ such that $P(A_n) = \frac{1}{7^n}$ for $n \in \mathbb N$. Determine the probability of the event $\...

1
2 3 4 5
8