Question

Doubt on a proof of Borel-Cantelli lemma

score 379 · Answer 1 · 2017-07-24 15:38:10

379

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Doubt on a proof of Borel-Cantelli lemma

Published on 24 Jul 2017 - 15:38

score 112 · Answer 2 · 2017-09-01 13:20:02

112

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a.s convergence and convergence in probability.

Published on 01 Sep 2017 - 13:20

score 144 · Answer 3 · 2017-09-18 19:26:18

144

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If $(X_i)$ is i.i.d. with finite third moment, then $n^{-1/3}\cdot\max\limits_{1\leq i\leq n}|X_i|\to0$ almost surely

Published on 18 Sep 2017 - 19:26

score 684 · Answer 4 · 2017-09-21 22:12:42

684

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Prove that a sum containing a Poisson variable is finite with probability 1

Published on 21 Sep 2017 - 22:12

score 495 · Answer 5 · 2017-09-22 03:55:32

495

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Correctly interpreting the Second Borel-Cantelli Lemma in a coin toss problem

Published on 22 Sep 2017 - 3:55

score 625 · Answer 6 · 2017-09-22 08:05:37

625

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Proof that $P(A_{n}\,\text{i.o.})=1$ implies $P(\cup_{i=1}^{\infty}A_{i})=1$

Published on 22 Sep 2017 - 8:05

score 242 · Answer 7 · 2017-09-24 09:43:52

242

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Showing that $ P\left(\lim\limits_{n \to \infty}\frac{S_n}{n} \text{ exists in } \mathbb R\right) = 0. $

Published on 24 Sep 2017 - 9:43

score 1k · Answer 8 · 2017-09-30 10:45:21

1k

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Second lemma of Borel-Cantelli: Normal distribution

Published on 30 Sep 2017 - 10:45

score 46 · Answer 9 · 2017-10-03 17:19:12

46

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Proving this statement without the independence condition (Borel 0-1 Law)

Published on 03 Oct 2017 - 17:19

score 780 · Answer 10 · 2017-10-13 16:45:16

780

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Almost sure convergence of sum of random variables

Published on 13 Oct 2017 - 16:45

score 238 · Answer 11 · 2017-10-13 16:51:48

238

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$c_n\to\infty$ so that $X_n/c_n\to 0$ a.s.

Published on 13 Oct 2017 - 16:51

score 326 · Answer 12 · 2017-10-15 17:46:19

326

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$P(A_n\space\text{i.o.})=0$ or $1$ according as $\sum_{n\geq 1}(1-F(\lambda_n))<\infty$ or $=\infty$

Published on 15 Oct 2017 - 17:46

score 149 · Answer 13 · 2017-10-19 08:25:00

149

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Existence of a Brownian motion - Proof.

Published on 19 Oct 2017 - 8:25

score 158 · Answer 14 · 2017-10-30 00:01:13

158

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How to apply the Borel-Cantelli lemma to the Hausdorff distance between a uniform sample on $[0,1]$ and the aforementioned interval

Published on 30 Oct 2017 - 0:01

score 350 · Answer 15 · 2017-11-07 17:52:10

350

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Exercise: almost sure convergence

Published on 07 Nov 2017 - 17:52

List Question

Doubt on a proof of Borel-Cantelli lemma

a.s convergence and convergence in probability.

If $(X_i)$ is i.i.d. with finite third moment, then $n^{-1/3}\cdot\max\limits_{1\leq i\leq n}|X_i|\to0$ almost surely

Prove that a sum containing a Poisson variable is finite with probability 1

Correctly interpreting the Second Borel-Cantelli Lemma in a coin toss problem

Proof that $P(A_{n}\,\text{i.o.})=1$ implies $P(\cup_{i=1}^{\infty}A_{i})=1$

Showing that $ P\left(\lim\limits_{n \to \infty}\frac{S_n}{n} \text{ exists in } \mathbb R\right) = 0. $

Second lemma of Borel-Cantelli: Normal distribution

Proving this statement without the independence condition (Borel 0-1 Law)

Almost sure convergence of sum of random variables

$c_n\to\infty$ so that $X_n/c_n\to 0$ a.s.

$P(A_n\space\text{i.o.})=0$ or $1$ according as $\sum_{n\geq 1}(1-F(\lambda_n))<\infty$ or $=\infty$

Existence of a Brownian motion - Proof.

How to apply the Borel-Cantelli lemma to the Hausdorff distance between a uniform sample on $[0,1]$ and the aforementioned interval

Exercise: almost sure convergence

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