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8 questions
1
vote
1
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185
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Sum of $X_k$ with $\mathbb{P}(X_k=\pm 1) = 1/2\pm 1/(2\sqrt{k})$
Let $\{X_k\}$ be a sequence of mutually independent random variables with
\begin{align}
\mathbb{P}(X_k = 1) & = \frac{1}{2} + \frac{1}{2\sqrt{k}},
\\
\mathbb{P}(X_k = -1) & = \frac{1}{2} - \...
- 269
6
votes
1
answer
168
views
Weak convergence of random measures generated by non-negative martingales?
If I have a sequence of non-negative continuous martingales $(M_n(x))_{n\ge 1}$ on $x\in[0,1]$, i.e. for each fixed $n$, $M_n:[0,1]\to[0,\infty)$ is a continuous process, and for each fixed $x\in[0,1]$...
- 3,808
1
vote
1
answer
83
views
Integral of $M^\text{*} - M$ with respect to $M^\text{*}$ is zero for $M^\text{*}$ the running maximum of $M$ a continuous local martingale
Given $M$ a continuous local martingale, and $M^\text{*} = \sup_{0 \leq s \leq t} M_s$ its running maximum, we consider the finite variation integral
$$
I_T:= \int_0^T (M^\text{*}_s - M_s) \, \text{d}...
- 3,808
1
vote
1
answer
140
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Does a sequence that verifies the assumptions of a square integrable martingale on some event need to be convergent on this event?
I came across this claim by reading some literature on stochastic approximation.
Let $(\Omega, \mathcal{A}, \mathbb{P}$) be a probability space, $(\mathcal{F}_n)$ a filtration on it. Let $(\epsilon_{n}...
- 1,172
0
votes
0
answers
71
views
Conditions for existence of a semi-martingale representing a system of probability measures
Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$.
Does there exist a semi-martingale $(X_t)_{t\in[0,1]}$ ...
- 5,405
5
votes
1
answer
652
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Proof of Pinelis (1992) - Banach space inequalities
I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3:
Let $(f_n)$ be a martingale in a separable ...
- 128k
4
votes
1
answer
302
views
Zero-one law for an independence-like structure
I am a number theorist by profession, so apologies if the answer to this question is "trivially true" or "trivially false".
Let $(\Omega, \mathcal{A}, P)$ be a (non-atomic) probability space. Let $(\...
- 29.8k
4
votes
1
answer
443
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Uniform martingale convergence of Radon-Nikodym derivatives of a convex set of probabilities
Cross posted at MSE here. I'm hoping someone here can help complete zhoraster's answer. Any hints or references are appreciated.
Let $(\Omega, \mathcal{F})$ be a measurable space equipped with a ...
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