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David Williams: "Probability with martingales" Kolmogorov 3-series theorem proof

score 366 · Answer 1 · 2026-03-27 00:58:42

366

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David Williams: "Probability with martingales" Kolmogorov 3-series theorem proof

Published on 27 Mar 2026 - 0:58

score 154 · Answer 2 · 2026-03-27 00:59:55

154

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If $E(X_n) = 1/n$ and $\mathrm{Var}(X_n) = 1/n^2$ then $X_n\to0$ almost surely

Published on 27 Mar 2026 - 0:59

score 79 · Answer 3 · 2017-06-10 11:37:42

79

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What's $\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$?

Published on 10 Jun 2017 - 11:37

score 981 · Answer 4 · 2017-06-12 20:38:17

981

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How to prove a sequence of functions converges uniformly to a function $f$ if and only if $\lim_{n\to\infty}\sup\{|f(x)-f_n(x)|:x\in S\}=0$?

Published on 12 Jun 2017 - 20:38

score 2.8k · Answer 5 · 2017-06-14 21:14:11

2.8k

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How to deal with lim sup and lim inf?

Published on 14 Jun 2017 - 21:14

score 3.9k · Answer 6 · 2017-06-22 15:04:54

3.9k

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Showing that the Fatou's lemma inequality can be strict.

Published on 22 Jun 2017 - 15:04

score 129 · Answer 7 · 2026-04-02 22:33:29

129

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Finding quotient norm

Published on 02 Apr 2026 - 22:33

score 74 · Answer 8 · 2017-07-01 11:33:06

74

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Proving inequality with $ \lim \inf$ in Sobolev space

Published on 01 Jul 2017 - 11:33

score 129 · Answer 9 · 2017-07-05 15:12:07

129

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How to show that $\limsup_{n\to \infty}\frac{a_n}{n}\leq \limsup_{n\to\infty}(a_{n}-a_{n-1})$ for any sequence $(a_n)$?

Published on 05 Jul 2017 - 15:12

score 214 · Answer 10 · 2017-07-10 20:03:41

214

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$u_n$ bounded, $v_n$ convergent then $\lim\sup(u_n + v_n) = \lim\sup (u_n) + \lim (v_n)$

Published on 10 Jul 2017 - 20:03

score 1.2k · Answer 11 · 2017-07-11 17:07:26

1.2k

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Proof of equivalence of two definitions $\limsup$ and $\liminf$

Published on 11 Jul 2017 - 17:07

score 334 · Answer 12 · 2017-07-18 19:26:03

334

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Confused by limit superior and limit inferior definition.

Published on 18 Jul 2017 - 19:26

score 67 · Answer 13 · 2017-07-19 09:59:52

67

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limsup of a functon and sequences

Published on 19 Jul 2017 - 9:59

score 56 · Answer 14 · 2026-03-28 21:06:54

56

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On $t\mapsto\max_{0\le s\le t}|f(s)|$ with $f$ $\alpha$-Holder function s.t. $f(0)=0$

Published on 28 Mar 2026 - 21:06

score 46 · Answer 15 · 2017-07-28 10:09:21

46

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Example of a sequence $(a_{n})_{n\in \mathbb{N}}$ such that $\liminf_{n\rightarrow \infty}a_{n}=-1$ and $\limsup_{n\rightarrow \infty}a_{n}=1$

Published on 28 Jul 2017 - 10:09

List Question

David Williams: "Probability with martingales" Kolmogorov 3-series theorem proof

If $E(X_n) = 1/n$ and $\mathrm{Var}(X_n) = 1/n^2$ then $X_n\to0$ almost surely

What's $\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$?

How to prove a sequence of functions converges uniformly to a function $f$ if and only if $\lim_{n\to\infty}\sup\{|f(x)-f_n(x)|:x\in S\}=0$?

How to deal with lim sup and lim inf?

Showing that the Fatou's lemma inequality can be strict.

Finding quotient norm

Proving inequality with $ \lim \inf$ in Sobolev space

How to show that $\limsup_{n\to \infty}\frac{a_n}{n}\leq \limsup_{n\to\infty}(a_{n}-a_{n-1})$ for any sequence $(a_n)$?

$u_n$ bounded, $v_n$ convergent then $\lim\sup(u_n + v_n) = \lim\sup (u_n) + \lim (v_n)$

Proof of equivalence of two definitions $\limsup$ and $\liminf$

Confused by limit superior and limit inferior definition.

limsup of a functon and sequences

On $t\mapsto\max_{0\le s\le t}|f(s)|$ with $f$ $\alpha$-Holder function s.t. $f(0)=0$

Example of a sequence $(a_{n})_{n\in \mathbb{N}}$ such that $\liminf_{n\rightarrow \infty}a_{n}=-1$ and $\limsup_{n\rightarrow \infty}a_{n}=1$

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