# Tagged Questions

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### Show $W_{\infty}$ is not tail measurable but {$W_{\infty}=0$} is

The random walk: ({$\omega_{n}$},$\mathbb{P}$) simple random walk on d-dimensional integer lattice $\mathbb{Z}^{d}$ and the random environment: $\eta$={$\eta(n,x):n\in\mathbb{N}, x\in \mathbb{Z}^{d}$} ...
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### Expectations of martingales

Consider a martingale $(M_n)_{n \geq 0}$ adapted to a filtration $(\mathcal F)_{n \geq 0}$ on a probability space $(\Omega, \mathcal F, P)$. Prove that, for each $k \leq n$; $$E(M_n M_k) = E(M_k^2)$$ ...
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### Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
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### Martingale Proofs

I havent been able to find an analogous question and our textbook is lacking in good examples, so I could use a little help with this rather straight forward martingale problem: Let X=(Xn) be a ...
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### Showing a function of random walk is a martingale

I would like a hint for the following problem: Consider a biased random walk on the integers with probability $p<1/2$ of moving to the right and probability $1-p$ of moving to the left. Let ...
I'm having trouble solving exercise 7.7.1 in Grimmett & Stirzaker's Probability and Random Processes, which reads: Let $X_1,X_2,\ldots$ be random variables such that the partial sums ...
Dear all, I hope you can help me with the proof of the following result: Fact If $X$ is a continuous local martingale, then $[X]_t < \infty$ a.s. for every $t \geq 0$, where $[X]$ denote the ...