# Tagged Questions

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### 1D biased random walk - is the event of infinte many returns a tail event?

I am considering a biased random walk: $X_1,X_2,\dots$ iid with $\mathbb{P}(X_1=1)=p$ and $\mathbb{P}(X_1=-1)=1-p$ with $p\in[0,1]\backslash\{1/2\}$, $S_n=X_1+\dots+X_n$. In this setting I want to ...
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### Uniform integrability of the maximum of a random walk with negative drift

Given $S_k^{(n)} = X_1^{(n)} + ... + X_k^{(n)}$ for all $k,n\in\mathbb{N}$, where the $X_i^{(n)}$'s are iid with mean $-\gamma$ for some $\gamma > 0$ and unit variance. Let ...
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### Recurrence for a random walk question

Let $X_i$'s be iid and define $X_1+\ldots+X_n=S_n$. I was trying to show that if $S_n$ is recurrent, then $S_{2n}$ is also recurrent. Assume these walks are in $\mathbb{R}^d$. Using Chung-Fuchs ...
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### Non - nearest neighbor random walk in $\mathbb{Z^{2}}$

$\textbf{Problem:}$ let {$X_{n} : n ≥ 0$} be any symmetric random walk on $\mathbb{Z^{2}}$ whose jumps have finite second moment. That is, $X_{0} = 0$ , {$X_{n} − X_{n−1} : n ≥ 1$} are mutually ...
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### Random walk on $\mathbb{Z}^d$

Problem Let $\{X_n\}_{n=0}^{\infty}$ be a random walk on $\mathbb{Z}^d$ such that; $X_0=(0,0,\cdots,0)$ and $\{X_n-X_{n-1}\}_{n=1}^{\infty}$ are mutually independent, identitically distributed ...
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### Find a asymptotic upper bound for $\sum_{n=N}^{\infty}p_{ii}^{(n)}$ for a asymetric one-dimensional simple random walk

For asymmetric one-dimensional simple random walk, that is $$P(X_n = X_{n-1} + 1) = p = 1 - P(X_n = X_{n-1} - 1)$$ for some $p \ne 1/2$, provide an asymptotic upper bound for ...
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### First step analysis on random walk

Let us consider random walk on integers {0,1,...,N} where $P(N,N)=1$,$P(0,1)=1$, $P(N,N-1)=0$ and all other connections have probability $\frac{1}{2}$. Using first step analysis, compute $p_{00}$ for ...
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### random walk in a certain environment

Consider the following random walk in one dimension, starting from $r(0)=0$. $$r(i+1) = r(i) + \xi,$$ where $\xi(i, r(i))$ is an increment with distribution $P(\xi=1) = \frac{c^{r(i)}}{i-r(i)+1}$ ...
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### ruin of the gambler with probability to die

Consider a random walk on $\mathbb{Z}$ starting from $i >0$. With probability $p$ it moves to the nearest neighbor on the left, with the same probability it moves to the nearest neighbor on the ...
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### mean displacement inequality for random walk with drift away from zero

Suppose $X_n$ is a nearest neighbor random walk on the integers with transition probabilities biased towards moving away from zero but with the bias asymptotically vanishing as you move away from ...
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### Number of times above a linear boundary for a finite variance random walk

I consider a random walk $(S_n)$ with mean zero and finite variance, and $\epsilon>0$. Is it true that $$\mathbb{E}\left[\sum_{n=0}^{+\infty} 1_{S_n>n\epsilon}\right] < +\infty \quad ?$$ ...
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### Stopping Time, Random Walk

I'm trying to solve this problem and don't know where to start. If someone could prove it or tell me how or point me to any relevant information I'd very much appreciate it. Let $(s_n)_{n\geq0}$ be a ...
Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is $$\nu \left( dx\right) = A ... 1answer 152 views ### Random walk with zero drift Let X_{k+1} = X_k+\xi_k be a random walk on \Bbb R starting from 0 and such that \mathsf E\xi_0 = 0, \mathsf{Var}[\xi_0]>0. Is that true that$$ -\infty = ...
This hints that $E(|S_n|)\,\!$, the expected translation distance after ''n'' steps, should be of the order of $\sqrt n$. In fact, \lim_{n\to\infty} \frac{E(|S_n|)}{\sqrt n}= \sqrt{\frac ...