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

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recurrence simple random walk in one dimension before hitting time

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
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35 views

Random walks with finite chance of escape

In a recent answer I gave a combinatorial interpretation for the sum $\sum_{n=1} \binom{2n}{n}\frac{4^{-n}}{n+1}=1$: namely, that it corresponded to the probability of all outcomes adding to $1$. A ...
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2answers
38 views

Limit value of a product martingale

This question came from a problem i was solving for self-study. I'll state the problem first: Let $Y_n \sim \mathcal N(0,\sigma^2)$ be independent normally distributed variables, $X_n = ...
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28 views

1-dimensional random walk with barrier

Let $X$ be a random walk on $\mathbb{Z}_{\ge 0}$ starting at $0$, with step size 1, and there is a barrier at 0 so that if one tries to move to -1 it stays at 0 (non-reflecting). If we fix the number ...
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52 views

Lattice Path Spaces.

It is well known that the number of paths from $(0,0)$ to $(n,k)$ in $\mathbb{N^2}$ with the set of steps $\{(1,0),(0,1)\}$ is ${n+k \choose k}$. This is the minimum number of steps needed to get to ...
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61 views

2D random walk variation

If a point on a 2D lattice is allowed to take a random walk by taking a unit step either up, down, left or right, there is probability $1$ of reaching any point (including the starting point) as the ...
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1answer
29 views

Random walk on free group on two elements

Let $F_2$ be the free group on two elements, generated by $\{a, b\}$. We perform a random walk on $F_2$, starting at the identity element $e$ and uniformly at random selecting one of ...
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1answer
59 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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25 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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37 views

Probability of Stopping Time Taking specific value - Random Walk 1d

We are considering a simple random walk $(X_n)_{n\in\mathbb{N}}$ starting at $X_0=0$ with $X_n=\sum_{i=1}^nY_i$ where $Y_i$ are iid and $\mathbb{P}(Y_i=1)=\mathbb{P}(Y_i=-1)=\frac{1}{2}$. We want to ...
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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
18 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
21 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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22 views

Weighted random walk in 1-dimension

Suppose we have random walker on a line, he can only stay on sites which are, say, a distance $a$ from each other. At each step he can go left or right. Every time he steps on a site, makes the ...
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59 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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47 views

Stationary distribution for random walks on directed graph

There is an equation (Eq. (2)) in reference by Lovasz and Winkler about the stationary distribution of a random walk on directed graphs that I would like to find references for where the equation is ...
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1answer
54 views

Random walk around circle

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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98 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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40 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
30 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
28 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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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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Estimation of random walk maximum and minimum positions

I am trying to prove that, if a simple and symmetric random walk $S$ starts at $S_0 = 0$ and finishes at $S_n = N$ with $N > 0$, then if there is a maximum $M > N$ and a minimum $B < 0$ (both ...
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69 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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39 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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31 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
64 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
46 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
61 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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56 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
56 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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42 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
58 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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Proof of “changes of sign” in one-dimensional random walk model [Feller's section 3.5, page 84]

Consider the one-dimensional random walk of a particle. We shall denote the individual steps by $X_1, X_2, \cdots$ with $X_i = \pm 1$ and the positions by $S_1, S_2, \cdots$ with $S_i = X_1 + X_2 + ...
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78 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
42 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
446 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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15 views

Writing a proof that a certain algorithm generates the correct transition matrix for a quantum walk?

Regarding quantum walks, I have a transition matrix $M$ and a particle vector $P$ and I have determined that the elements of $M$ have to be positioned in a certain way so that the position of the ...
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40 views

Expected number of steps in a random graph walk

Suppose I have a directed graph $D(V, A)$ where the edges have weights on them. Let's notate the weight function $w: A \rightarrow [0, 1]$. If $f, t \in V$ and $a \in A$ such that $a = (f, t)$ then ...
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1answer
67 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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87 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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253 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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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
39 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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39 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
50 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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33 views

Random walk on an infinite resistive lattice

I have been referring to a paper http://arxiv.org/abs/physics/0405135 to determine the effective resistance using random walks for an infinite square resistive lattice Though the author seems to ...
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69 views

proving null recurrence of random walk (Markov chain)

How would I prove that the zero state of a random walk with a positive probability of staying in the same state is null recurrent. (sorry if this isn't a random walk and just a Markov chain.) eg. ...
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39 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
48 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 ...