Question

limsup and liminf of sequence of random variables and Borel Cantelli

score 1.7k · Answer 1 · 2015-10-11 21:38:23

1.7k

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limsup and liminf of sequence of random variables and Borel Cantelli

Published on 11 Oct 2015 - 21:38

score 1.1k · Answer 2 · 2026-03-21 07:38:59

1.1k

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Given $\{A_n\}$, with $P\left(A_n\right)\to 1$, there exists a subsequence $\{n_k\}_{k\ge 1} \;$ such that $\; P\left(\bigcap_k A_{n_k}\right) > 0$

Published on 21 Mar 2026 - 7:38

score 1.3k · Answer 3 · 2015-10-31 19:20:02

1.3k

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What is the probability of randomly choosing a rational number from $[0,1]$

Published on 31 Oct 2015 - 19:20

score 786 · Answer 4 · 2026-03-18 21:16:31

786

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Is it correct to say that ($\color{red}{(} \limsup |W_k|/k\color{red}{)} \le 1) \supseteq \limsup \color{red}{(}|W_k|/k \le 1\color{red}{)}$?

Published on 18 Mar 2026 - 21:16

score 384 · Answer 5 · 2026-03-26 09:07:21

384

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Probabilistic proof for $\prod_{n} (1-p_n) > 0$ (Considering the $p_n$'s as parameters of random variables and using Borel-Cantelli and continuity)

Published on 26 Mar 2026 - 9:07

score 118 · Answer 6 · 2015-11-12 19:01:49

118

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Find $P(\limsup A_n)$ given $P(A_{n+1}) \ge \frac{n}{n+1} P(A_n)$

Published on 12 Nov 2015 - 19:01

score 186 · Answer 7 · 2015-11-16 16:03:07

186

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Let $X_1,X_2,\cdots$ be i.i.d $p^{\text{th}}$ integrable random variables. Show that $\cfrac{1}{n^{1/p}} \sup_{k\le n} |X_k| \to 0$ almost surely.

Published on 16 Nov 2015 - 16:03

score 138 · Answer 8 · 2015-11-19 23:01:43

138

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Prove $\liminf(M_n = -1) \subseteq ((\lim \frac{S_n}{n}) = -1)$

Published on 19 Nov 2015 - 23:01

score 671 · Answer 9 · 2026-03-02 21:53:42

671

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Proof convergence of random variables (almost sure)

Published on 02 Mar 2026 - 21:53

score 482 · Answer 10 · 2015-11-20 11:04:17

482

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Concept of "eventually almost surely" as an artefact of measure-theoretic axioms?

Published on 20 Nov 2015 - 11:04

score 213 · Answer 11 · 2026-03-13 09:45:15

213

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Clarification on Borell-Cantelli lemma

Published on 13 Mar 2026 - 9:45

score 38 · Answer 12 · 2015-11-21 14:15:52

38

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Prove that $\liminf [X_n = 2]) \subseteq ([\lim S_n] = \infty)$

Published on 21 Nov 2015 - 14:15

score 268 · Answer 13 · 2015-11-23 10:16:05

268

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Lebesgue measure of closure of a set

Published on 23 Nov 2015 - 10:16

score 112 · Answer 14 · 2015-11-23 21:08:34

112

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Probability of a independent sequence of random variables

Published on 23 Nov 2015 - 21:08

score 78 · Answer 15 · 2015-11-23 21:37:41

78

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Probability of an sequence of independent random 2-dimensional vectors

Published on 23 Nov 2015 - 21:37

List Question

limsup and liminf of sequence of random variables and Borel Cantelli

Given $\{A_n\}$, with $P\left(A_n\right)\to 1$, there exists a subsequence $\{n_k\}_{k\ge 1} \;$ such that $\; P\left(\bigcap_k A_{n_k}\right) > 0$

What is the probability of randomly choosing a rational number from $[0,1]$

Is it correct to say that ($\color{red}{(} \limsup |W_k|/k\color{red}{)} \le 1) \supseteq \limsup \color{red}{(}|W_k|/k \le 1\color{red}{)}$?

Probabilistic proof for $\prod_{n} (1-p_n) > 0$ (Considering the $p_n$'s as parameters of random variables and using Borel-Cantelli and continuity)

Find $P(\limsup A_n)$ given $P(A_{n+1}) \ge \frac{n}{n+1} P(A_n)$

Let $X_1,X_2,\cdots$ be i.i.d $p^{\text{th}}$ integrable random variables. Show that $\cfrac{1}{n^{1/p}} \sup_{k\le n} |X_k| \to 0$ almost surely.

Prove $\liminf(M_n = -1) \subseteq ((\lim \frac{S_n}{n}) = -1)$

Proof convergence of random variables (almost sure)

Concept of "eventually almost surely" as an artefact of measure-theoretic axioms?

Clarification on Borell-Cantelli lemma

Prove that $\liminf [X_n = 2]) \subseteq ([\lim S_n] = \infty)$

Lebesgue measure of closure of a set

Probability of a independent sequence of random variables

Probability of an sequence of independent random 2-dimensional vectors

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