List questions from borel-cantelli-lemmas | MathQa

limsup and liminf of sequence of random variables and Borel Cantelli

Published on 12 Jan 2025 - 8:38

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 12 Jan 2025 - 8:53

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

Published on 12 Jan 2025 - 8:53

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 12 Jan 2025 - 8:53

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 12 Jan 2025 - 8:52

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

Published on 12 Jan 2025 - 8:53

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 12 Jan 2025 - 8:53

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

Published on 12 Jan 2025 - 8:53

Proof convergence of random variables (almost sure)

Published on 12 Jan 2025 - 8:53

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

Published on 12 Jan 2025 - 8:53