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

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9
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113 views

Expected range of simple random walk in $\mathbb{Z^2}$

Let $(Y_k)_{k\geq0}$ be a simple random walk process. The range of an $n$-step random walk, $R_n$, is a random variable that characterizes the number of distinct points visited at time $n$: ...
7
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275 views

Is there a connection between the 3D random walk constant and the partition function?

In thinking about this question, I took a look at Pólya's random walk constants and was struck by the fact that an expression for the constant for a three-dimensional random walk, ...
5
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30 views

Random Walk Without Repetitions

Suppose that we simulated a random walk on $\mathbb Z$ starting at $0$. At each step, we transition from position $x$ to position $x-3,\,x-2,\,x-1,\,x+1,\,x+2,$ or $x+3$ with equal probability. If ...
5
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45 views

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) = ...
5
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158 views

Intuition for the optimality of bold play

There is a standard result (I think originally by Dubins and Savage) that if one wants to maximise the probability of winning a certain amount in an unfair game of chance then an optimal strategy is ...
5
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221 views

Where does directed random walk hit the boundary?

I have a problem that I more or less know the answer to, but would really like to see it done in a systematic, rather than ad hoc way. In spite of this, I will pose the question in a very concrete ...
4
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94 views

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 ...
4
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58 views

Standard deviation of a quantum walk?

The standard deviation of a classical random walk with $n$ steps is $\sqrt n$ - Standard deviation of a random walk. I have read in many places that the standard deviation of a quantum walk $n$ with a ...
4
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227 views

Random walk on $\mathbb{Z}$ with more than two possible steps

Let be $\{X_n\}_{n\in \mathbb{N}}$ random walk on $\mathbb{Z}$. Let be $$P(X_{n+1} = k + a| X_n = k)= p_a$$ for $a\in \mathcal{A} \subset \mathbb{Z}$. Let say that $X_0 = 0$. I am interested in ...
4
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65 views

random walk on real line

Suppose I start at $A>0$ and every period I either move a distance $B$ to the right with probability $p$ or a distance $C$ to the left with probability $1-p$. The expected move is positive: ...
4
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144 views

Conditional probability and integrating out part of a random walk

Suppose that I have a random walk process defined by $\alpha_{t+1}$ ~ N$(\alpha_t, \omega^2)$. Given $\alpha_t$ and $\alpha_{t+2}$, I understand why the conditional formula for ...
4
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131 views

probability of this event happening

Play $(n+1)t$ rounds of the same coin-tossing game and the coin is fair ($n$ is a fixed natural number). Please help me find the following probability: $P$(the number of rounds of tossing that ...
4
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103 views

Completeness of random walks in multiple dimensions?

I was reading Artificial Intelligence: Modern Approach (Norvig and Russell), and there was a footnote that really caught my attention. I apologize if the problem is more in the domain of CS than ...
3
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36 views

Random walks and their uses

Can anyone provide some motivation behind the use of random walks? I know they're used a lot in computer science, in things like page walk (I think that's what it was called- something like pagerank), ...
3
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55 views

Simple random walk conditioning on non-return

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0=0$. Let $k$ and $j$ be two positive integers. Let $P_{k,j}$ be the probability that the walker hits the vertex ...
3
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38 views

Self-avoiding random walk on $\mathbb{Z}^2$ getting stuck

Let $W_n$ be a self-avoiding random walk (SAW) on $\mathbb{Z}^2$, starting at the origin. Formally, $W_0=0$ and for $n\ge 0$, $W_{n+1}$ is chosen uniformly from the neighbours of $W_n$ which were not ...
3
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74 views

Sum of sequence of random variables infinitely often positive

Let $X_1,X_2,\ldots$ be an infinite sequence of independent (but not necessarily identically distributed) random variables with $E(X_i)=0$ for all $i$. Set $S_n=\sum_{i=1}^n X_i$. I want to show that ...
3
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74 views

Expected minimum of a finite random walk.

So I couldn't find any resource for how to calculate the expected minimum of a random walk. Since it is such the minimum of the random variables are actually not independent as they are cumulative ...
3
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34 views

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 $$ \mathbb{E}\left[\sum_{n=0}^{+\infty} 1_{S_n>n\epsilon}\right] < +\infty \quad ? $$ ...
3
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54 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
3
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205 views

Two gamblers' ruin

I'm trying to work out the solution to a variant of the gambler's ruin. Here's my version: There are two very unlucky but friendly gamblers A and B who decide to pool their money together to form a ...
3
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69 views

Diffusion on a graph and its dual

Is there a relation between the diffusion of a random walker on a planar graph and that on the dual of the graph? It seems perhaps intuitive that if the diffusion on the graph is slow (in comparison ...
2
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33 views

An inequality for symmetric random walk

I need to show that if $(X_j)$ are symmetric i.i.d. random variables with partial sums $S_n:= \sum_{j=1}^n X_j$, then for all $x \geq 0$ $$P(|S_n| > x) \geq \frac{1}{2} P(\max_{1 \leq j \leq n} ...
2
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45 views

2d random walk on the nonnegative quadrant using martingale techniques

I know the basics of (discrete time) martingales, and I'd appreciate any help and suggestions on how to prove the following using martingale techniques. Let $Z_n$, $n\ge 0$ be a random walk on the ...
2
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21 views

Estimating the discrete laplacian to prove recurrence of simple random walk for d=2

Given a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}$ we define the discrete laplacian of $f$, $\triangle_df$, by the following rule $\triangle_df(x,y)= \dfrac{f(x + 1, y)+f(x, y + ...
2
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34 views

Sum of $\{X_n\}$ iid random variables contained in a compact interval implies each $X_i=0$ a.s.?

I am working through an exercise that starts with a sequence i.i.d. random variables where for $a\leq b$, $$\Pr\left(\lim\sup_n \sum_{i=1}^{n} X_i \in [a,b] \right) \neq 0.$$ Does this require $X_i ...
2
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30 views

Comparing hitting time of two random walks

There are two random walks, $S^t_i=S^{t-1}_i+ X_i^t$ for $i=1,2$, $X^t_i$ i.i.d they have boundaries $h_1$ and $h_2$ respectively. I'm wondering if it's possible to calculate the probability that one ...
2
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72 views

Hitting time of a maximum of random walk converges to that of Brownian motion

Suppose $S_n$ is a simple random walk; formally, $S_n=\sum_{i=1}^n X_i$ for $X_i\sim\mathcal{U}(-1,1)$, i.i.d.. Denote by $M_n$ the maximum of the random walk on $n$ steps; formally, $M_n=\max_{0\le ...
2
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44 views

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 ...
2
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0answers
47 views

The number of paths, which touches or crosses the abscissa

If $S_n$ is a random walk s.t. $S_0=1$. $S_n=X_1+X_2+...+X_n$ for $n\ge 1$ and for any $i\in N$ $P[X_i=1]=P[X_i=-1]=1/2$ for $r\ge 1$ calculate the number of paths from time $0$ to $2n-1$ ...
2
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71 views

Random Walk Return Probabilities – Is there an intuition to understand them?

Every mathematician is familiar with the result (due to Pólya) that for a random walk in a $d$-dimensional lattice, the probability $p(d)$ for returning to the origin eventually is $1$ if $d=1,2$, but ...
2
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53 views

A random walk on the unit distance graph in $\mathbb{R}^n$

Define a graph $G_n$ whose vertices are the points in $\mathbb{R}^n$ with an edge connecting any two points that are one unit apart. Such a graph is called the unit distance graph in $\mathbb{R}^n$. ...
2
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66 views

Random walk with $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} < \infty$

Consider a random walk started at $S_0=0$, denoted $S_n = \sum_{k=1}^{n}X_k$, where $X_1$, $X_2$... are the i.i.d increments. If we have $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} ...
2
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46 views

Expectation of a Random Walk

I am researching Random Walks and trying to find how to get their expectations. I have studied Markov chains before. I have found one way of getting the expected number of steps to reach a state by ...
2
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0answers
59 views

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 ...
2
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0answers
146 views

Finding functions where the increase over a random interval is Poisson distributed

I'm trying to construct a type of function $f(t_1, t_2)$ that counts the number of deterministically simulated Poisson events between two points in time. We can use a single valued function ...
2
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0answers
42 views

Sums of independent random variables

I was unsuccessful in deriving a good estimate of the distance below. Let $(X_{n})_{n \geqslant 1}$ be a sequence of i.i.d. random variables, and let $(\varepsilon_{n})_{n\geqslant 1}$ be a sequence ...
2
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0answers
99 views

random walk with possibility to freeze

Consider a Random Walk on a one-dimensional lattice. The walker starts moving at time $0$ from $x=0$. At every step, the walker moves to the right with probability $p$, to the left with probability ...
2
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0answers
64 views

Random walk type problem with time increments

Imagine you have $\$50$ and every $2$ minutes you either gain or lose $33$ cents. How would you model the evolution of the hypothetical bankroll for the next hour? My approach based on what i've read ...
2
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0answers
62 views

Dimension free Concentration bounds for Martingales

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
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274 views

Teleporting random walk

Given a directed graph $G = (V,E)$, to define a random walk on $G$ with a transition probability matrix $P$ such that it has a unique stationary distribution (as mentioned in this paper) I used a one ...
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418 views

Random walk, Cat and mouse

Here is the problem. In graph G, on different vertices there is cat and mouse. Cat and mouse do independent random walk, but time is synchronous, in one unit of time both cat and mouse do one step. ...
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0answers
115 views

Spectrum of Transition Matrix for Random Walk

Consider the symmetric random walk on $\{0, 1, \dots, n\}$ with transition probabilities $P(j \to j \pm 1) = 1/2$ for $1 \le j \le n-1$ and $P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2$. ...
2
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100 views

Equilibrium distributions for a finite urn scheme

Given an urn with $n$ (fixed) balls that can be red or black, and given two parameters $0 < p, \, q < 1$, keep doing the following: Flip a $p$-coin. If heads come up, remove a black ball or ...
2
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0answers
126 views

Repeated reflection

This is a problem from Feller's book introduction to probability theory and its application, Vol 1, Chap 3 problem 3. Let $a$ and $b$ be positive, and $-b <c<a$. Prove the number of paths to ...
2
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0answers
280 views

Probability distribution of a self avoiding walk

Preliminary: Consider a walk on the lattice $\mathbb{Z}_d$ lattice of length $N$. In a normal random walk, if we let $N$ get large the end position has a probability distribution (PDF) that looks ...
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25 views

Reading material on Random Walk on S_n using Transpositions

I am from an engineering background and I wanted to get hold of some very basic reading material on Random Walk on $S_n$ (symmetric group on n letters) using Transpositions. Could someone suggest ...
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0answers
30 views

Expected time of reaching 0 of a simple symmetric random walk

Consider the symmetric, simple random walk on $S = \{0, 1, \ldots , k\}$ for $k \in \mathbb N$. Let $$T = \min \{ n \in \mathbb N_0|X_n = 0\}$$ be the first time where the process reaches $0$ and ...
1
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0answers
29 views

capacity of biased random walk in $\mathbb{Z}^2$

Let $P_{x,y}$ the probability that a random walk starting from $x$ will ever visit $y$. Consider a biased random walk in $\mathbb{Z}^2$. Let $A_k$ be the set of vertices having a distance less than ...
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0answers
17 views

a problem on random walk and maximu of its summation

Suppose $X_n$ is random walk with $P(X_n=1)=1-P(X_n=-1)=p=1-q$. $M_n=\max_{1\le i\le n} S_i$, $Y_n=M_n-S_n, T_a=\min\{S_n=a\}$. Find $P(\max_{0\le k\le T_a} Y_k <y).$