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

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Why random walk sample path seen as if it is continuous-time stochastic process?

Random walk is a discrete-time stochastic process. In many references, instead of using dots to draw its sample path, why does random walk use line-styled graph as if it is a continuous-time ...
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52 views

Random walk on a tree

Consider a Cayley tree with coordination number 3 (http://en.wikipedia.org/wiki/Bethe_lattice). Consider two sites, $x$ and $y$, having a distance $k$ one from another. What is the probability that ...
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58 views

Maximum of *Absolute Value* of a Random Walk

Suppose that $S_{n}$ is a simple random walk started from $S_{0}=0$. Denote $M_{n}^{*}$ to be the maximum absolute value of the walk in the first $n$ steps, i.e., $M_{n}^{*}=\max_{k\leq ...
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17 views

Let $S_n$ be a Simple Random Walk. What is $E[S_m|S_n]$ if $m < n$?

Let $S_n = W_1 + ... + W_n$ be a simple random walk with $W_i$ IID and $P[W_i = 1] = P[W_i = -1] = 1/2$. Find $E[S_m | S_n]$ when (a) $m > n$ and (b) $m < n$. For part (a), I get the answer of ...
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61 views

Expected number of steps

I play a game as follows. A bucket contains four red balls and three green balls. At each step, a ball is chosen at random from the bucket, with each of the balls there being equally likely to be ...
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2answers
106 views

Bounding profits of gambler by Azuma Inequality

A gambler plays the following game: In each round, he can pay any $0 < p < 1$ dollars, and win 1 dollar with probability p (independently). Show that the probability that the gambler's net ...
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1answer
161 views

Random Surfer as a Markov Chain

Consider a random surfer who begins at a web page (a node of the web graph) and executes a random walk on the Web as follows. At each time step, the surfer proceeds from his current page A to a ...
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109 views

Deducing results about continuous time random walks from the corresponding discrete time result

Is there any standard way to prove results about continuous time random walks from the corresponding results for discrete time random walks? Specifically, my problem is that I was reading Lawler and ...
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32 views

Cicurlar random-walk.

Suppose you have a computer network with 5 code as following. Packet can arrive at any node and any other node can be its destination equal uniform probability. Determine the average number of ...
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48 views

Random-walk in a pentacle (5 nodes)

There are a total of 5 nodes at the edge of a pentagram At each node, you have a 4 choices which will lead you to either a destination node or non-destination node. Assume the decision of path is ...
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105 views

Couple/Compare two stochastic processes and prove an intuitive proposition

Consider a stochastic process (denoted $X$) $X_0, X_1, X_2, \ldots$ (not necessarily a Markov Chain) over state space $\{0, 1, \cdots, n \}$. The transition probabilities are ($n$ is the sink state) ...
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46 views

Simple Random Walk; Proof hitting theorem; Ballot Theorem

Suppose that $(X_{n}:n\in\mathbb{N})$ is a $\pm1\mbox{-valued sequence.}$ Let $p\in(0,1)$ and $p=\mathbb{P}(X_{i}=1)\mbox{ and}\mathbb{P}(X_{i}=-1)=1-p=q$ . Define the simple random walk ...
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69 views

Probability of going to the origin in a random walk

Been given this as practice for my Stochastic Processes course. I'm fairly new to the concept, so I haven't been exposed to a general method. Any hints/tips for the following? A gambler plays a ...
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1answer
91 views

Duration of a Gambler's Ruin game against an opponent with infinite credit

A gambler enters the casino with $x\$$ in his pocket and sits on some table. At each iteration he bets $1\$$ and either wins $1\$$ with probability $p\leq\frac{1}{2}$ or loses $1\$$. Assuming that ...
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147 views

Showing that lim sup of sum of iid binary variables $X_i$ with $P[X_i = 1] = P[X_i = -1] = 1/2$ is a.s. infinite

Let $(X_i)_{i\in\mathbb{N}}$ be an i.i.d. sequence of binary random variables with $$P[X_i = 1]=P[X_i = -1] = \frac{1}{2}$$ and let $$S_n = \sum_{i=1}^{n} X_i.$$ I'd like to show that $$P[\lim ...
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101 views

Probability- Coin Flipping Game

If you play a game where you flip a coin if it lands heads you win £1 and tails you lose £1.If you start with $£K$ what is the probability that you are bankrupt after $n$ games? MY ATTEMPT I ...
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94 views

Expected number of moves of a random walk inside a simplex

Few days ago, Roger Blazey, retired head maths teacher, at LinkedIn group told about the following problem: “This problem was raised as part of a lecture at The Biennial AAMT (Australian ...
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55 views

Bound on sum random variables and Martingales

Suppose $X_n=q$ with probability $p$, and $X_n=-p$ with probability $q$ where $p+q=1$. Prove that for every $n$, the probability that $S_k\geq b$ for any $k$ as $1\leq k \leq n$ is at most ...
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1answer
268 views

Expectation of hitting time for simple symmetric random walk

Assume there is a simple symmetric random walk $$S_n=X_1+...+X_n,\quad S_0=0$$ where $\mathbb P(X_i=\pm 1)=\frac{1}{2}$. Define $T=\inf\{n:S_n=1\}$. How to compute $\mathbb E(T)$? My idea: if ...
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51 views

Maximum time-to-exit of random walk in R^n

I am trying to solve the following problem : Given a set $A$ in $\mathbb{R}^n$ and a point $p$ , I want to find a convex subset of $A$, call it $C$, such that $p$ is in $C$ and random walk starting at ...
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2answers
167 views

better expression for simple random walk

Let $P_{k,j}$ be the probability that a simple symmetric random walk starting from the origin reaches the point $k \in \mathbb{N}$ precisely in $j$ steps without ever returning to the origin. ...
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1answer
106 views

A question about independence of sigma algebras (generated by random variables)

Let $X_1, X_2, \ldots$ i.i.d random variables. Is it possible that $$\{X_{n+1} \in B\} \in \sigma({X_1, \ldots, X_n})$$ for some $B$? Why yes/not? I want to show that $\sigma(X_{n+1})$ and ...
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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 + ...
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81 views

continuous random walks, wiener process, ito process: “snowballing” for high enough volatility?

I'm finishing a project for my ODE class and ran into some strange behavior involving a SDE (not exactly sure how to say this, but...) generated by an Ito process, using the Wiener process. I guess ...
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76 views

Brownian motion is almost surely continuous

Why is Brownian motion required to be almost surely continuous instead of merely continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
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1answer
37 views

What are the assumptions for applying Wald's equation with a stopping time

I am trying to understand the assumptions under which I am allowed to apply Wald's equation for a sum of a random number $N$ of random variables $X_n$, $1\leq n\leq N$. There seem to be several ...
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1answer
32 views

Expectation of constraint random walk

Problem description I am currently dealing with a practical problem that can be simplified to something like this: I start by setting a value to 0 Every minute, I try to increase or decrease 1 or ...
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37 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), ...
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87 views

Random walk on the real line

A particle stands on the origin of real line. In every second it jumps one unit to the left or right by the probability $\frac{1}{2}$. what is a probability that it reach to point $+1$?
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82 views

random walk on finite cyclic group

Suppose that I have a random walk on the finite cyclic group of order $d > 2$, where the initial probability distribution $Q$ assigns the values $p, q, r$ to $-1, 0, 1$, respectively, where $p + q ...
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1answer
55 views

Simple Random Walk and $n$th zero hitting time

I am reading an example in Durrett's book regarding the $n$th time the random walk hits 0. Consider a simple random walk, $X_i=1$ or $X_i = -1$ with equal probability. Let $S_n = X_1 + \dots + X_n$. ...
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30 views

Power law in power spectrum and memory.

If we generate white noise and do the FFT of it, we get the same amplitude for each of the frequencies. Therefore, the output of the FFT of the noise follows approximately the power law ...
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68 views

Random Walk on Clock Hands

We do a random walk on a clock. Each step the hour hand moves clockwise or counterclockwise each with probability 1/2 independently of previous steps. If you start at 1 what is the expected number ...
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90 views

How to analyse a random walk with random transition probabilities

Consider a $1$-dimensional random walk with discrete time steps. We start at the origin and at each integer position there is possibly different probability of moving right one step, or left one step. ...
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54 views

Simple Markov Chain: Random Walk on $\mathbb{Z}$

We are given a random walk on $\mathbb{Z}$, where $p_{i, i+1}= p < \frac{1}{2}$ and $p_{i,i-1}=1-p > \frac{1}{2}$, starting at $0$. Now we have to compute the probability that we eventually ...
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Simple Random Walk on Integers

Question concerning a simple random walk on 1D. Why the probability of hitting $\pm 2^n$ before return to $0$ is $2^{-n}$? I have no idea how to start... Thank's!
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26 views

Expected Value of Continuous Random Walk

I'm currently attempting my MMath master project. However, i'm a little stuck on an expected value of the continuous distribution. Its where i wish to change my random walk from a continuous walk to ...
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41 views

Random walk return for subgraph

Assume that $G$ is a finite graph and we have a simple random walk starting at some vertex $v$ of $G$. We fix $n$, and consider the probability that the random walk does not return to $v$ after $n$ ...
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12 views

Stochastic Process with mean reverting property

Here I am seeking for a definition of what kind of stochastic processes are called mean reverting stochastic process. That is, what are the properties that a stochastic process should obey in order to ...
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81 views

Random Walk, Markov Process

I'm stuck on a homework question and am wondering if anyone can offer some hints. Suppose we have some straight line graph G over the set $ V = \{1, 2, 3, ... , n\} $ of vertices, with an edge between ...
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32 views

Random walk on non-negative integers

Consider the Random walk on the non-negative integers with transition probabilities $$ p_{0,1}=1,~~~p_{i,i+1}=1-r,~~~p_{i,i-1}=r,~~~i\geq 1. $$ Determine $p_{00}^{(n)}$ As far as ...
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2answers
46 views

Expected time to get x units away when only able to move 1 unit either way

I know this is a common problem, but this problem has been bugging me after someone asked me it, and I can't find the answer anywhere on the Internet. Say we have a number line, and we start at the ...
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269 views

Random walk on infinite binary tree (recurrence, transience)

Consider a random walk on the infinite binary tree with root $x$ which has the following transition probabilities. $$ p_{x,0}=p_{x,1}=\frac{1}{2},~~~p_{y,y0}=p,~~~p_{y,y1}=q,~~\text{and ...
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Prove that $\text{Var} \tau = \frac{1 − (p − q)^2}{(p-q)^3} $ where $\tau$-stopping time

Let $S_n = \xi_1 + \dots + \xi_n$ be asimetric random walk such that $P(\xi_i = 1) = p > \frac{1}{2}$ and $P(\xi_i = -1) = q $. Let $\sigma^2 =1-(p-q)^2$ and let $X_n=(S_n-n-(p-q)n)^2 - \sigma^2n $ ...
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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 ...
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68 views

Guarantee to lose in +EV gamble?

I've checked it in numerical experiments but found it counter intuitive. A player start with $a_0>0$ dollars, let's denote the amount of money after $n$ rounds $a_n$ dollars. In each round, the ...
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1answer
91 views

Standard deviation of absolute distance of a 1D random walk

Given a 1D random walk (simple +1, -1 movements from the axis) I've seen proofs that the expected absolute distance tends to Sqrt(2*n/PI) and I've plotted graphs of 1D random walks along with this ...
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43 views

Boundary conditions for random walk

Consider a simple asymmetric random walk $S_n$ which goes up with probability $p$ and down with $1-p$. For $b<x<a$ let $$r(x) = P( S_n\text{ hits }a \text{ before }b |S_0 = x). $$ This ...
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33 views

Limit of a random walk

Suppose we have a simple random walk: $X_n\pm1$ with equal probabilities. For any finite $n$, $E[\sum_{k=1}^nX_k]=0$. Does it imply that $E[\lim_{n\rightarrow \infty}\sum_{k=1}^nX_k]=0?$ Thank's! ...
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21 views

Simple random walk in the limit

Consider simple random walk: $X_n=\pm1$ with equal probabilities. $S_n =\sum_{i=1}^nX_i$. For finite $n$ we can write $$S_n=\sum_{i=1}^nX_i=\sum_{i=1}^nX_i^+ -\sum_{i=1}^nX_i^-$$ So that ...