Question

Let $Y_n$ be i.i.d with $EY_n = 1, P(Y_n = 1) < 1, X_n = \prod_{k=1}^n Y_k$. Show martingale convergence of $X_n \to 0$ a.s.

score 779 · Answer 1 · 2022-02-18 21:13:51

779

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Let $Y_n$ be i.i.d with $EY_n = 1, P(Y_n = 1) < 1, X_n = \prod_{k=1}^n Y_k$. Show martingale convergence of $X_n \to 0$ a.s.

Published on 18 Feb 2022 - 21:13

score 40 · Answer 2 · 2022-02-19 06:10:27

40

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Find an expression for $X_n = E[Y | \mathcal{F}_n]$ with $\mathcal{F}_n = \sigma(\{1\}, \{2\}, \ldots, \{n-1\}, \{n, n+1, \ldots\})$

Published on 19 Feb 2022 - 6:10

score 194 · Answer 3 · 2026-03-31 14:29:09

194

Views

With $Y_n$ i.i.d, $EY_n = 1$, $X_n = \prod_{k=1}^n Y_k$, use Strong Law of Large Numbers to show $\frac{\log(X_n)}{n} \to c < 0$

Published on 31 Mar 2026 - 14:29

score 155 · Answer 4 · 2026-03-28 04:35:50

155

Views

What went wrong in my application of the Martingale Central Limit Theorem to a degenerate U-statistic?

Published on 28 Mar 2026 - 4:35

score 203 · Answer 5 · 2026-04-12 07:01:11

203

Views

Why do we require a stochastic integral to be a martingale?

Published on 12 Apr 2026 - 7:01

score 255 · Answer 6 · 2026-04-03 13:55:32

255

Views

A Càdlàg non-negative submartingale is class DL

Published on 03 Apr 2026 - 13:55

score 198 · Answer 7 · 2026-04-12 07:16:03

198

Views

Challenge: a Bit-flipping Game

Published on 12 Apr 2026 - 7:16

score 200 · Answer 8 · 2026-04-13 12:39:01

200

Views

moment inequality of martingales

Published on 13 Apr 2026 - 12:39

score 122 · Answer 9 · 2026-04-14 00:07:36

122

Views

How to show that $\sup_n E(X_n^2) < \infty$ for a certain martingale?

Published on 14 Apr 2026 - 0:07

score 42 · Answer 10 · 2022-03-02 12:38:04

42

Views

If a theorem is true for a sub/super-martingale does it also hold for a martingale?

Published on 02 Mar 2022 - 12:38

score 60 · Answer 11 · 2026-03-27 04:21:57

60

Views

How to show $\mathbb{E}(Y_{n+1}^{2} | \mathcal{F}_{n}) = \mu^{2} Y_{n}^{2} + \sigma^{2} Y_{n}$? (Unanswered repost)

Published on 27 Mar 2026 - 4:21

score 49 · Answer 12 · 2026-04-03 14:02:29

49

Views

Showing that $P(T_{-a}<T_b)=\frac{e^{2\mu b}}{e^{2\mu(a+b)}-1}$

Published on 03 Apr 2026 - 14:02

score 95 · Answer 13 · 2026-04-13 12:50:04

95

Views

How to show that $\{X_n\}$ is martingale?

Published on 13 Apr 2026 - 12:50

score 204 · Answer 14 · 2026-04-12 20:59:05

204

Views

Proof explanation of Durrett's probability theory theorem 4.3.1

Published on 12 Apr 2026 - 20:59

score 75 · Answer 15 · 2022-03-08 16:01:30

75

Views

If $M_t= M_0 + \int_0^t Y_s d B_s$ is an $L^2$ martingale on $[0,1]$, why is $E\left[\int_0^1 Y_s^2 ds\right] = E[M_1 B_1 - M_0 B_0]$

Published on 08 Mar 2022 - 16:01

List Question

Let $Y_n$ be i.i.d with $EY_n = 1, P(Y_n = 1) < 1, X_n = \prod_{k=1}^n Y_k$. Show martingale convergence of $X_n \to 0$ a.s.

Find an expression for $X_n = E[Y | \mathcal{F}_n]$ with $\mathcal{F}_n = \sigma(\{1\}, \{2\}, \ldots, \{n-1\}, \{n, n+1, \ldots\})$

With $Y_n$ i.i.d, $EY_n = 1$, $X_n = \prod_{k=1}^n Y_k$, use Strong Law of Large Numbers to show $\frac{\log(X_n)}{n} \to c < 0$

What went wrong in my application of the Martingale Central Limit Theorem to a degenerate U-statistic?

Why do we require a stochastic integral to be a martingale?

A Càdlàg non-negative submartingale is class DL

Challenge: a Bit-flipping Game

moment inequality of martingales

How to show that $\sup_n E(X_n^2) < \infty$ for a certain martingale?

If a theorem is true for a sub/super-martingale does it also hold for a martingale?

How to show $\mathbb{E}(Y_{n+1}^{2} | \mathcal{F}_{n}) = \mu^{2} Y_{n}^{2} + \sigma^{2} Y_{n}$? (Unanswered repost)

Showing that $P(T_{-a}<T_b)=\frac{e^{2\mu b}}{e^{2\mu(a+b)}-1}$

How to show that $\{X_n\}$ is martingale?

Proof explanation of Durrett's probability theory theorem 4.3.1

If $M_t= M_0 + \int_0^t Y_s d B_s$ is an $L^2$ martingale on $[0,1]$, why is $E\left[\int_0^1 Y_s^2 ds\right] = E[M_1 B_1 - M_0 B_0]$

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