# Tagged Questions

For questions on random walks, a mathematical formalization of a path that consists of a succession of random steps.

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### Probability on entering direction of a simple random walk

Let $X(n)$ be a simple random walk on $\Bbb{Z}^2$. Also we define $S_{R} = \inf\{n > 0 : X(n) \notin [-R, R]^2 \}$ : the exit time of the square $[-R, R]^2$, $T_{v} = \inf\{n > 0 : X(n) = v\}$...
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### Test a law-of-iterated-logarithm-like result, with numerical simulation

I have a non-standard random walk $S_n$ for which the increments are not exactly independent (I could describe it, but it would be a totally different long and complex topic). I expect it to have ...
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### Let $\{ X_{n}\} _{n\geq1}$ be IID s.t $\mathbb{E}[X_{i}]=0$ and $|X_{i}|\leq K$. Show $S_{n}$ visits $[-K,K]$ infinitely often.

Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of IID random-variables s.t $\mathbb{E}\left[X_{i}\right]=0$ and $\left|X_{i}\right|\leq K$ . Let $S_{n}=\sum_{i=1}^{n}X_{i}$ , I want to show ...
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### Theory of random walk

Let 0 < p < 1 and let $S_n$ be the simple random walk with step probabilities p, 1 − p. In other words $S_n = X_1 + · · · + X_n$ and the {$X_i$} are i.i.d. random variables with distribution ...
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### Almost sure convergence of a martingale

I just learned martingales (with no depth) and I am working on the following question. Suppose $S_n$ is a a random walk on the integers and at each step, it increases by 1 with probability $p$ or ...
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### Gambler's Ruin with no set target for win

I have been presented with the following probability question: A compulsive gambler is never satisﬁed. At each stage he wins $€1$ with probability $p$ and loses $€1$ otherwise. Find the probability ...
How to prove that $\Bbb P_1(\lim\sup_{n\to\infty}\frac{\log \bf{x}(n)}{\log n}\le \frac{1}{2})=1$, with the suggestion that $\bf{x}(n)$ goes to $\infty$ like $\sqrt{n}$. Here $\bf{x}_n$ is the ...