# 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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### (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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### 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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### 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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### 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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### 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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### 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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### 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 ...
1 vote
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### 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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### 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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### 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 ...
### 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 $X_n \to X$ ...