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

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1answer
250 views

Proving a property of hitting times of a simple random walk on $\mathbb{Z}$

I'm reading the course notes of a probability course about martingales currently and I'm trying to solve some of the exercises, however I'm very much stuck with the following exercise: Let $\left\{ ...
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1answer
122 views

Are random walk variations Markov-Chains?

Let $S_{n}:= S_0 + \sum_{i=1}^{n}X_i$ be a simple random walk, $X_i$ are independent random variables with $P[X_i=1] = p, P[X_i = -1] = 1-p$. Let $M_n:=\max\{S_0, \dots, S_n\}$. The task at hand is ...
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58 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 ...
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1answer
162 views

Writing down the transition matrix of a discrete Markov chain

Please consider the following scenario: One person is walking along a discrete circle induced by $\mathbb{Z}/n\mathbb{Z}$ In each round we roll a dice with $w\in\left\{2,\ldots n\right\}$ sides If ...
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1answer
71 views

Simple Random Walk probability of first visit

Consider a particle that moves according to a simple random walk. Denote by $X_n$ the position of the particle immediately after step $n$. Assume that $X_0 = 0$ and that, at each step, the ...
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0answers
101 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 ...
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1answer
527 views

Random walk around circle [duplicate]

For one of the exercises of my homework I need to answer the following question, but I am not sure how I should apply gamblers ruin theory to solve this problem (it is stated as a hint, not that I ...
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1answer
263 views

Probability of random walk returning to 0

Given a symmetric 1-dimensional random walk starting at 0 -- what is the probability of the walk returning $k$ times to 0 after $2N$ steps? I know that the total number of paths it can take is ...
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0answers
157 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 ...
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1answer
99 views

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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1answer
55 views

Randomly moving about a lattice, probability that I return to my original location?

I thought of this question when I was walking aimlessly around my neighborhood. Here's my question: My house is on an infinite lattice of points: say my house is at $(0,0)$. I start walking north (it ...
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0answers
113 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 ...
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2answers
131 views

Random walk on tree

You begin at a root node that has 2 children. Each of those two children have two more children, and each of those children have two final children (i.e., there are 15 nodes in the graph). How do I ...
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0answers
79 views

Random looking Gray Codes or Hamiltonian Cycles on Hypercubes

Cyclic Gray codes come in many flavors and correspond 1-1 to Hamiltonian cycles on hypercubes. I would like to find a type that looks like a random walk on the hypercube. In a sense this is an ...
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0answers
62 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 ...
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2answers
201 views

Power series convergence of random walk transition matrix

I would like to find out if $$ \sum_{t=0}^\infty P^t = \left( I- P \right)^{-1} ~,$$ where $P = D^{-1}W ~ $ is a random walk transition matrix. $W$ is a matrix describing weights in a graph and ...
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1answer
129 views

{Probability}: choosing keys from a pool without replacement

The OP is trying to understand the following question. The OP understand that if you can always write out the term $$P(X=k) \implies (1-\frac{1}{N})(1-\frac{1}{N-1})\cdots(1-\frac{1}{N-k+1}),$$ ...
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1answer
148 views

Expected number of steps and probability

I have a problem that I am not quite sure how to solve using my elementary knowledge of probability. My question is this: suppose a friend and I are playing a game. We both start at 0 points, and ...
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1answer
63 views

Probability: deviation from the mean

I am having trouble to understand the following. If $S_n=X_1+X_2+......+X_n$, where X_1,X_2 are Bernouli (p). I don't understand this. So you get an intermediate point Constant* sqrt(n). To the ...
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1answer
152 views

rate of convergence of absorbing markov chain

Let $G$ be a biconnected and non-bipartite graph. I can simulate a random walk on this graph with a markov chain. The stochastic matrix is $M = AD^{-1}$, where $A$ is the adjacency matrix of $G$ and ...
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0answers
93 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 ...
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1answer
76 views

Infinite series containing binomial coefficients

I've encountered the following series: $$\sum_{t=1}^\infty {1 \over 2^{t}}\, {{\large t} \choose {\large{t + x \over 2}}}$$ Is this series even convergent? I'm really lacking knowledge on series ...
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1answer
154 views

Function of a uniformly distributed continuous random variable

Basically, I'd like to add $n$ random vectors in a 2 dimensional space of unit length and of angle $\theta$ relative to a global axis. The probability density function of the angle $\theta$ is a ...
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1answer
116 views

Explanation on one-dimensional random walk in Feller's book

Consider the random walk on the integer number line, $\mathbb{Z}$, which starts at 0 and at each step moves $+1$ or $−1$ with equal probability. The probability for the event that "the first return to ...
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3answers
705 views

Probability a random walk is back at the origin

I have a symmetric random walk that starts at the origin. With probability $1/6$ it goes right by one and with probability $1/6$ it goes left by one. With probability $4/6$ it stays put. After $n$ ...
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1answer
98 views

Solving a recurrence for a random walk revisited

I previously asked about the following recurrence which I get when trying to solve a random walk problem given a positive integer $x$. $p_i = \dfrac{p_{i-1}}{2} + \dfrac{p_{i+2}}{2}$ if $0< i < ...
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1answer
136 views

Recurrence for random walk

I have the following recurrence which I get when trying to solve a random walk problem given a positive integer $x$. $p_i = \dfrac{p_{i-1}}{2} + \dfrac{p_{i+2}}{2}$ if $0< i < x$ $p_i = 1$ if ...
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2answers
1k views

Simulating Diffusion/Wiener Process with Random Walk

I hope this is the right section for this kind of questions. I am trying to simulate, with MATLAB, a diffusion model starting from a Random Walk. I am using a Random Walk with information increment ...
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0answers
57 views

Random walk on a graph

For a random walk say from point $x$ to $y$ on a graph, How is the probability of a Random walker reaching point $y$ before returning to $x$ related to the expected of the number of visits to point ...
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1answer
102 views

Reaching a level before another for a random walk

Suppose we are given a simple random walk starting in $0$, i.e. $(X_k)_{k\in\mathbb{N}}$ with $P[X_k=+1]=P[X_k=-1]=\frac{1}{2}$. What is the probability of hitting the level $a$ before hitting the ...
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0answers
51 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 $2n-1$ ...
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1answer
196 views

Probability of random walk traversal

Consider a random walk on an connected, non-bipartite, undirected graph G. Show that, in the long run, the walk will traverse each edge with equal probability. Note: The walk can traverse each edge ...
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0answers
137 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 ...
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2answers
60 views

Generate random sample with three-state Markov chain

I have a Markov chain with the transition matrix $$\pmatrix{0 & 0.7 & 0.3 \\ 0.8 & 0 & 0.2 \\ 0.6 & 0.4 & 0}$$ and I would like to generate a random sequence between the three ...
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1answer
23 views

Please explain $E[S_{min(n,T)} ]= E [S_{0}]=0$

If $S_{n}$ is a simple random walk i.e $X_{k}= +/- 1$ with prob = 0.5 T = inf {n > = 0 |$S_{n}$ = 1} is a stopping time. T is finite almost surely. .Explain $E[S_{min(n,T)} ]= E [S_{0}]=0$ I know ...
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2answers
231 views

Urn Problem - black and white balls, infinite trials.

Imagine that there are 10 black balls and 20 white balls in an urn. Two balls are removed at random from the urn. The second ball removed is recolored, such that it matches the color of the first ...
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0answers
61 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 $\mathbb{R}^n$. ...
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0answers
72 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 \} ...
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0answers
70 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 ...
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57 views

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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1answer
218 views

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 ...
0
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1answer
32 views

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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1answer
22 views

Estimating a point on graph from multiple random values.

I am developing a mobile game app that needs to find a point on a map based on a set of observed values. The app allows users to touch points on a map to the closest proximity of where they think an ...
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0answers
102 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 ...
2
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1answer
407 views

Simple random walk expectation problem

Consider a simple symmetric random walk, $\left\{X_n\right\}$, where the following hold; $$X_n = X_{n-1} + Y_n$$ for $n = 1, 2, 3,\ldots$ $$Y_1, Y_2, Y_3, \ldots \text{iid }, \mathbb{P}(Y_1 = 1) = ...
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0answers
110 views

Hitting time Distribution of a Gaussian Random Walk

I am trying to find out the exponential decay rate of the Probability $Pr(T>n)$ where $T$ is the first hitting time of a gaussian random walk with i.i.d random variables i.e. ...
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1answer
69 views

Bounds on median of random walks

If $k$ random $n$-step $\pm 1$ walks start at 0, and the $i$th walk ends at position $X_i$, how big is $\text{median}_i \, |X_i|$? Is there a bound along the lines of $\text{P}(\text{median}_i \, ...
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3answers
482 views

Distribution of the first passage time of a Gaussian random walk

Does anyone know the distribution for the first passage time of a Gaussian random walk i.e. $$ S_n = \sum_{i=1}^n X_i $$ where $X_i$ are iid normally distributed random variables. The first passage ...
3
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1answer
109 views

Random walk where increment depend on current position

Consider the following stochastic process, $$b(i+1) = b(i) + \xi_i (b_i),$$ where $\xi_i(b_i) \in \{-1, k \}$ are the independent increments having the following distribution: $$\begin{align} P (\xi ...
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1answer
102 views

2 dimensional random walk - hit of targets

Consider a random walk in $\mathbb{Z}^2$, $x(j) = x(j-1) + \xi_j$, where the increments are random variables independent and identically distributed with finite support, the expectation $m := ...