1
vote
0answers
15 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 ...
1
vote
1answer
37 views

Invariant mesure of a reflected random walk

Let $(X_n), n \geq 0$ be a Reflected Random Walk defined by: $X_0 = 0$ and: $ X_{n+1}=\max( 0 , X_n + \xi )$ $\xi $ is a random variable such that $P(\xi=a)=\theta$ and $P(\xi=-b)=1-\theta$ for a ...
0
votes
0answers
30 views

Reflected random walk

Suppose that $X_n$ is a reflected (in 0) random walk with parameter $\theta$. So $X_{n+1}-X_n = 1$ with probability $\theta$ , and -1 with probability $1-\theta$ when $X_n \geq 1$, if $X_n=0$ then ...
1
vote
1answer
60 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\{ ...
0
votes
1answer
29 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 ...
0
votes
1answer
24 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 ...
0
votes
1answer
24 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 ...
0
votes
0answers
50 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 ...
0
votes
0answers
33 views

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 ...
2
votes
0answers
42 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 ...
0
votes
1answer
17 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 ...
2
votes
0answers
59 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 \} ...
0
votes
0answers
46 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. ...
1
vote
0answers
25 views

Prove equilibrium theorem without irreducibility and aperiodicity

I have to solve the following question: Consider a random walk Markov chain on $S = \{1, 2, \ldots, 100\}$. If the chain is between 2 and 99, it selects one of the adjacent states with equal ...
0
votes
0answers
50 views

Absorption probability in 1D RW with asymmetric step sizes, $ x<0 $

What is the probability of absorption at $ 0 $, as a function of position $ x $, for a 1D random walk (on $ \mathbb{Z} $) with asymmetric step sizes? For example, suppose that you can take two steps ...
0
votes
1answer
142 views

Probability for asymmetric random walk

How to express this in equation form(in terms of position(x) and time(N)), like the one for symmetric random walk, $\displaystyle P(x,N) = \frac{N!}{(\frac{N+x}{2})! ...
5
votes
1answer
178 views

The random walk of two drunks

The problem is such: two drunks start at either end of an alleyway of length n. Apart from at the ends, they each move one step forwards or one step backwards randomly. At the ends of the alley they ...
0
votes
0answers
43 views

Expected hitting time of one barriers

I have learned the skill to deal with Expected hitting time of one of two barriers. Now I have a similar one.The walker starts from x=0 and the barriers are located in x=+n.The walker can move one ...
2
votes
0answers
49 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 ...
3
votes
1answer
139 views

What is the probability a random walk hits x before it hits y?

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
3
votes
0answers
33 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 $$ \mathbb{E}\left[\sum_{n=0}^{+\infty} 1_{S_n>n\epsilon}\right] < +\infty \quad ? $$ ...
2
votes
1answer
69 views

How to model a stochastic process, continuous in stepsize, which converges against a simple random walk?

I want to compute the probability distribution for a stochastic process with discrete number of steps, where each real value has a nonvanishing probability to be the next stepsize. And I want to ...
0
votes
0answers
26 views

Why must a stochastic process be at least second order in terms of differential equations?

A first order differential equation in $q(t)$ has a unique path through each possible value of $q(0)$. This is opposed to a stochastic process (e.g. random walk), where any place might be "hopped ...
0
votes
1answer
71 views

First-passage probability with absorbing boundary at origin (No Laplace)

I have the following problem which I would like to solve without using Laplace transform. Can you possibly help or provide pointers? What is the first-passage probability, and mean first-passage time ...
2
votes
2answers
72 views

Random walk on one-dimensional lattice - understanding the expression $pe^{i\theta} + qe^{-i\theta}$

I've started reading the book - First Steps in Random Walks and in the very first example in Chapter 1 they talk about a random walk on a one-dimensional lattice. If we consider a particle starting ...
1
vote
0answers
47 views

Random Walk in confined region and loop configurations

Suppose I take a random walk on a 2 dimensional square lattice, but this lattice plane has a finite size, e.g. Dx*Dy. I can not cross the boundary, my step length is the lattice cell size, I either go ...
0
votes
0answers
35 views

How to perform a stochastic search of the locality of a node in a network?

In a graph that may be a random graph (ER graph), scale free network, etc. I would like to obtain a distribution of the locality of the nodes surrounding a ...
1
vote
0answers
112 views

Bayesian random walk

Suppose that, at first, I am trying to estimate the mean and standard deviation of some data that I assume to be normally distributed. My prior is gaussian with mean $\mu_0$ and variance $\sigma^2_0$. ...
0
votes
0answers
94 views

On the second derivative of Wiener process

As we all know, continuous white noise is the derivative, with respect to time, of a Wiener process. My question is that does the second derivative of Wiener process exists? If so, what is it and how ...
2
votes
1answer
97 views

Computation of a mean (random sum)

Let $X_1$, $X_2$, ... be independent and identially distributed positive random variables and define the sum $S_n = X_1 + X_2 + ... + X_n$. Consider the first time $N$ where $S_N \ge b$ with a given ...
2
votes
1answer
76 views

A continuous random walk of length 1

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity? For one ...
4
votes
1answer
425 views

Random walk as a martingale?

Let $S_0$, $Z_1$, $Z_2$, $\ldots$ be independent random variables. $S_n=S_0+Z_1+\cdots+Z_n$, $n=0,1,2,\ldots$ $S_n$ is a random walk starting in a random point, $S_0$ I need to find out, when it is a ...
1
vote
1answer
132 views

Random walk probability non-symmetric steps

I currently have a probability class tutorial question that I have no idea where to begin. At first instinct, I thought it may have been a CTMC question or branching question, but now I have no idea, ...
1
vote
1answer
203 views

A Boundary crossing result for discrete brownian bridge

Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process $$ ...
4
votes
2answers
117 views

Prove that a random walk on $\mathbb{Z}_+\cup \{0\}$ is transient

Prove that a random walk on $\mathbb{Z}_+ \cup \{0\}$ is transient with $p_{i,i+1}=\frac{i^2+2i+1}{2i^2+2i+1}$ and $p_{i,i-1}=\frac{i^2}{2i^2+2i+1}$. So since this Markov chain has only a single ...
0
votes
2answers
269 views

The variance of a simple random walk/process

I've been trying to wrap my head around this for the past day. Please help! Let $\epsilon_i = \pm 1$ with equal probabilities independently for $i=1,...,N$. Then $Z_i = \epsilon_1 + ... + \epsilon_i$ ...
0
votes
1answer
116 views

The random walk $S_n=a+\sum_{i=1}^nX_i$

Consider a variant of random walk defined as $$S_n=a+\sum_{i=1}^nX_i,$$ where $X_i$ takes either value $2$ with prob= $p$ or value $-1$ with prob =$1-p$. What is $P(S_n=b)$?
1
vote
1answer
58 views

Inequality between two Random Walks

Let's consider two Random Walks, $$x^{(1)}_t = x_0 + \sum_{i=1}^{t}\xi^{(1)}_i,$$ $$x^{(2)}_t = x_0 + \sum_{i=1}^{t}\xi^{(2)}_i.$$ The random variables $\xi^{(1)}_i$ are i. i. d. They take values on ...
3
votes
1answer
144 views

A Coupled Random Walk on the xy-Plane

Consider a point on the $xy$-plane whose position is updated in iterations. In each iteration, the point undergoes, with equal probability, either an $A$- or a $B$-update, defined as follows: ...
0
votes
1answer
413 views

Expected hitting time of one of two barriers

In the webpage "hitting time of one of two barriers", the probability that a non symmetric random walk hits one of two barriers is computed. The walker starts from $x=0$ and the barriers are located ...
0
votes
2answers
208 views

hitting time of one of two barriers

Let's consider a one-dimensional Random Walk. At each time the walker moves of one step to the right with probability $p$ and to the left with probability $q$, with $p+q=1$. The walk is not symmetric, ...
1
vote
0answers
231 views

Extinction probability of a simple birth death process

X is a simple birth death process with birth rate $\lambda n$ and death rate $\mu n$ Embedded within a simple birth death process is a simple random walk. Let $Y_n $ be the value of X at the time of ...
1
vote
1answer
118 views

Finding the exact solution of a difference equation

We know a particle moves two units to the right with probability $p$, or $1$ unit to the left with probability $q$, hence $(p+q=1)$. $$q_k=P\left(S_n=0\mid S_0=k\right)$$ We are asked to find the ...
1
vote
1answer
129 views

Show that $Z_t = Z_0 \exp\left( \mu t + X_t \right)$ is well defined where $X_t$ is a Jump process (Lévy process)

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 ...
3
votes
2answers
1k views

Null-recurrence of a random walk

In a random walk on $\mathbb{Z}$ starting at $0$, with probability 1/3 we go +2, with probability 2/3 we go -1. Please prove that all states in this Markov Chain are null-recurrent. Thoughts: it is ...
0
votes
1answer
153 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 = ...
1
vote
2answers
129 views

Using random walks to predict behavior rather than matrix decomposition

I want to create a model that tries to predict a user's behavior based on the random walks of similar users. The problem is similar to Netflix's recommendation challenge. One of the popular solutions ...
4
votes
1answer
350 views

Covariance of Brownian-motion-like processes

We know that $\operatorname{Cov}(B_s,B_t)=\min(s,t)$ if $B_t$ is Brownian motion. What is $\operatorname{Cov}(B_{f(s)},B_{f(t)})$ for some injective $f$? How can I write $B_{f(t)}$ in an Ito ...
0
votes
1answer
158 views

mean reverting random walks

I need a set of stochastic processes $x_i(t)$ with the following characteristics: At each time $t$, the jump of each variable can be just $+s$ or $-s$; The processes have to be mean reverting, so ...
1
vote
1answer
147 views

Increasing entropy of random walk in regular graph

Let $P$ be a transition matrix of a random walk in an undirected regular graph $G$. Let $\pi$ be a distribution on $V(G)$. The Shannon entropy of $\pi$ is defined by $$H(\pi)=-\sum_{v \in ...