List questions from borel-cantelli-lemmas

Published on 24 Nov 2015 - 10:36

88

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Estimate the probability $P(X > C\frac{(n-1)^p}{\sqrt{n}})$ for $X\sim N(0,1), C>0, p>1/2$

Published on 25 Nov 2015 - 20:37

1.3k

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Almost Surely convergence with Bernoulli

Published on 28 Nov 2015 - 16:46

69

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Deterministicaly stops winning, yet if conditioned on time there is positive probability to win in the future.

Published on 29 Nov 2015 - 2:52

1.6k

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If an event $\{X_n = 1\}$ happens infinitely often with probability 1, why can't $X_n \overset{a.s.}\to 1$?

Published on 29 Nov 2015 - 3:58

503

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Recurrent random walk

Published on 01 Dec 2015 - 14:02

218

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How to apply law of large numbers for this problem

Published on 03 Dec 2015 - 11:03

622

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Show that $\limsup_{n\rightarrow\infty} \frac{\sum_{k=1}^{n} X_k}{n}<\infty$ a.s.

Published on 08 Dec 2015 - 19:59

149

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Asymptotic estimate of $\binom nk$

Published on 09 Dec 2015 - 4:56

110

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Prove the convergence of a random variable

Published on 11 Dec 2015 - 8:24

321

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Let $X_1, X_2, \ldots$ be independent r.v.'s with $0 \leq X_n \leq 1$ and $\sum_n E(X_n) = \infty$. Show $\sum_n X_n = \infty$ with probability 1?

Published on 14 Dec 2015 - 14:33

44

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Show something converge to infinity strongly

Published on 15 Dec 2015 - 17:40

937

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The infinite sum of integral of positive function is bounded so function tends to 0

Published on 15 Dec 2015 - 20:07

47

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Constants $a_{n}$ s.t. $a_{n}(max_{1\leq k\leq n}X_{k}-a_{n})\stackrel{d}{\to}L$; $X_{k}\sim N(0,1)$

Published on 15 Dec 2015 - 22:28

129

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Is this sigma-additivity or union bound?

Published on 20 Dec 2015 - 8:20

List Question

Using Borel-Cantelli Lemma to show the almost sure divergence of $S_n/n$

Estimate the probability $P(X > C\frac{(n-1)^p}{\sqrt{n}})$ for $X\sim N(0,1), C>0, p>1/2$

Almost Surely convergence with Bernoulli

Deterministicaly stops winning, yet if conditioned on time there is positive probability to win in the future.

If an event $\{X_n = 1\}$ happens infinitely often with probability 1, why can't $X_n \overset{a.s.}\to 1$?

Recurrent random walk

How to apply law of large numbers for this problem

Show that $\limsup_{n\rightarrow\infty} \frac{\sum_{k=1}^{n} X_k}{n}<\infty$ a.s.

Asymptotic estimate of $\binom nk$

Prove the convergence of a random variable

Let $X_1, X_2, \ldots$ be independent r.v.'s with $0 \leq X_n \leq 1$ and $\sum_n E(X_n) = \infty$. Show $\sum_n X_n = \infty$ with probability 1?

Show something converge to infinity strongly

The infinite sum of integral of positive function is bounded so function tends to 0

Constants $a_{n}$ s.t. $a_{n}(max_{1\leq k\leq n}X_{k}-a_{n})\stackrel{d}{\to}L$; $X_{k}\sim N(0,1)$

Is this sigma-additivity or union bound?

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