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

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42
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7answers
11k views

Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
7
votes
1answer
2k views

Mean distance from origin after $N$ equal steps of Random-Walk in a $d$-dimensional space.

I am looking for a formula that evaluates the mean distance from origin after $N$ equal steps of Random-Walk in a $d$-dimensional space. Such a formula was given by "Henry" to a question by "Diego" ...
15
votes
1answer
8k views

Expected Value of Random Walk

Can someone very simply explain to me how to compute the expected distance from the origin for a random walk in $1D, 2D$, and $3D$? I've seen several sources online stating that the expected distance ...
15
votes
2answers
2k views

Random walk on $n$-cycle

For a graph $G$, let $W$ be the (random) vertex occupied at the first time the random walk has visited every vertex. That is, $W$ is the last new vertex to be visited by the random walk. Prove the ...
3
votes
3answers
4k views

biased random walk on line

Lets say we start at point 1. Each successive point you have a, say, 2/3 chance of increasing your position by 1 and a 1/3 chance of decreasing your position by 1. The walk ends when you reach 0. ...
2
votes
1answer
602 views

Probability of asymmetric random walk returning to the origin

Consider the random walk $S_n$ given by $ S_{n+1} = \left\{ \begin{array}{lr} S_n+2 & with & probability & p\\ S_n - 1 & with & probability & 1-p \end{array} ...
1
vote
1answer
515 views

Symmetric random walk with bounds

can anyone help me with this: We are considering a symmetric random walk that ends if level 3 is reached or level -1 is reached. Start=0 What is the expected number of walks? So I am looking for: ...
7
votes
3answers
701 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 $n$ ...
3
votes
1answer
441 views

Random walk $< 0$

Suppose ${X_t}$ is a random walk with mean zero. (either discrete or continuous time) Fix a time $T$. What is: $P[X_t < 0 \text{ for all } t \leq T]$? In words, what's the probability the random ...
0
votes
1answer
58 views

Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
0
votes
1answer
103 views

probability, random walk, Markov chain question

Let $P$ be a transition matrix for a regular Markov chain and let $w$ be it’s equilibrium vector. Show that $w$ has no zero entries.
5
votes
3answers
5k views

1D random walk-probability to go back to origin

Suppose There is a random walk starting in origin while probability to move right is 1/3 and probability to move left 2/3.What is the probability to return to the origin. Thank you
1
vote
1answer
225 views

Returning Paths on Cubic Graphs

Suppose we have a 3-edge-colorable cubic graph with $N$ vertices. How many paths of length $N$ exist that return to its origin? Or putting it differently: What is "Pólya's Random Walk Constant" on ...
5
votes
1answer
748 views

Probability distribution for the position of a biased random walker on the positive integers

I initialize a biased one-dimensional random walk on the positive integers at the origin, $x = 0$, which also serves as a reflecting boundary blocking steps onto the negative integers. Let's say that ...
4
votes
1answer
306 views

Conditional return time of simple random walk

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$. Call $\tau_k = \min\{t \in \mathbb{N}\, : \, \, S_t =k \}$, the hitting time of $k \in \mathbb{N}$. Call $\tau^* = \min\{t ...
10
votes
1answer
505 views

Is there an intuitive way to see this property of random walks?

For an $n$-step symmetric simple random walk (start at origin 0 and each step 1 unit towards left or right with equal probability,) an interesting fact is that the probability that you stop exactly at ...
4
votes
1answer
652 views

Expectation of $TS_T$ where $T$ is the absorption time at $\{a,-a\}$ of a simple symmetric random walk $\{S_n\}$

I was trying to calculate the expectation of $T^2$ using some martingale and got that I needed the expectation of $TS_T$. Any idea?
2
votes
2answers
332 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 = ...
1
vote
1answer
125 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, \ldots \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
1
vote
2answers
96 views

Showing stopping is finite almost surely

Consider a discrete random walk taking values +1 or -1 with probabilities p and q, respectively. Let $S_n = \sum_{k=1}^{n}X_k$. Let $[-A,B]$ be an interval, $A,B \geq 1$. Now define $$\tau =\min(n:n ...
0
votes
1answer
24 views

Random Walk Limit Behavior

Suppose $\{ X_t \}$ is a sequence of i.i.d. random variables, with support $\{-1,1\}$ and distribution $P(1)=P(-1)=1/2$. Thus, $S_t = \sum_{s=1}^{t} X_s$ is a zero mean random walk. Also, $S_t$ is a ...
5
votes
1answer
1k views

Expected number of steps in a random walk with a boundary

Let's say I am trying to climb a flight of $N$ stairs. Each time I want to take a step, I flip a fair coin. Heads means I take a step up; tails means I take a step down. If I'm at the bottom of the ...
3
votes
4answers
715 views

Random walking and the expected value

I was asked this question at an interview, and I didn't know how to solve it. Was curious if anyone could help me. Lets say we have a square, with vertex's 1234. I can randomly walk to each ...
2
votes
1answer
2k views

Expected number of steps till a random walk hits a or -b. [duplicate]

On wikipedia I read that the expected number of steps till a 1D simple random walk hits either $a$ or $-b$ is equal to $ab$. (I have seen this result also on other websites.) However, no proof or ...
0
votes
1answer
71 views

Symmetric Random Walk / Prove $S = \inf\{n : X_n = 7\}$ and $T = 10^{12} \wedge S$ are $\{\mathscr F_n^Y\}$-stopping times.

Given a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $\mathscr F_n = \mathscr F_n^Y$, let $Y_1, Y_2, ...$ be iid random variables w/ $P(Y_n = ...
0
votes
2answers
132 views

Non-Probabilistic Argument for Divergence of the Simple Random Walk

The simple random walk is one starting at $0$ with steps of $-1$ and $1$ with equal probability. Is there a proof not involving (too much) probability - preferably number-theoretic - of why this walk ...
0
votes
2answers
350 views

How to check that a sequence of numbers is random? [duplicate]

I have a sequence of numbers like 1,7,22,45,12,96,21,45,65,36,85,14,51,16,18,17,16....65... IS there any formula to check whether the sequence is random or not ? In my case odd numbers are ...
49
votes
1answer
2k views

Identity for simple 1D random walk

The question is to find a purely probabilistic proof of the following identity, valid for every integer $n\geqslant1$, where $(S_n)_{n\geqslant0}$ denotes a standard simple random walk: $$ ...
10
votes
2answers
2k views

Biased Random Walk and PDF of Time of First Return

I have a random walk process where each step the probability of $+1$ is $p$ and $-1$ is $q$, with $p+q=1$. $p$ may not equal $q$. The walker starts at zero. I want to know the probability that the ...
12
votes
3answers
415 views

Random walk on natural number

Problem: You are standing at the position $0$ on the line of natural numbers $0, 1, 2, ..., n$. From this position you go to $1$ with probability $1$, but from any other position $i$ you go to $i+1$ ...
8
votes
6answers
363 views

simplify summation of factorial (random walk)

I suspect that the expression $$\sum_{n=0}^N \frac{(N-2n)^2}{n!(N-n)!}$$ simplifies to $$\frac{2^N}{(N-1)!}$$ But I cannot find the intermediate steps. Can someone give me a hint how I can deduce ...
4
votes
1answer
66 views

Numerical evidence of law of iterated logarithm (random walk)

The law of iterated logarithm states that for a random walk $$S_n = X_1 + X_2 + ... X_n$$ with $X_i$ independent random variables such that $P(X_i = 1) = P(X_i = 1) = 1/2$, we have $$\limsup_{n ...
4
votes
1answer
196 views

Expected number of returns to zero in a symmetric random walk - closed form

The expected number of returns of a symmetric random walk is given by $\sum_{k=0}^n \binom{2k}{k} / 2^{2k} -1$ The exercise is to compute an explicit form for this. I tried to do this in the ...
2
votes
1answer
131 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 ...
1
vote
3answers
1k views

Exact probability of collision of two independent random walkers after N steps

Two drunks start together at the origin at $t=0$ and every second they move with equal probability either to the right or to the left, each drunk independently from the other. What is the probability ...
8
votes
1answer
158 views

How long until everyone has been in the lead?

Earlier, I asked a question about a series of competitions: A series of matches are held between n identical competitors. Each is won by one of the n with equal probability (no ties). I'm looking ...
5
votes
1answer
135 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 ...
4
votes
1answer
92 views

Show that $P(T \le n + N \mid \mathscr F_n) > \epsilon$ where T is a stopping time

Given random variables $Y_1, Y_2, \ldots \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
4
votes
1answer
459 views

Random Walk on Z

Let $S_n$ be the symmetric random walk on $\mathbb{Z}$. How do i calculate $P(\limsup_{n\rightarrow\infty} S_n=\infty)$? I already know that the probability is 1 but I don't really know how to start? ...
3
votes
1answer
227 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 ...
2
votes
3answers
145 views

Random walk problem in the plane

Let a particle in the plane $R^2$ executes random jumps at discrete times $t= 1, 2, ...$. At each step, the particle jumps from the point it is a distance of lenght one. The angle of any new jump ...
2
votes
2answers
2k views

What are some martingales for asymmetric random walks?

Here are some examples for symmetric ones: http://mathoverflow.net/questions/55092/martingales-in-both-discrete-and-continuous-setting/55101#55101 Is there a similar list for asymmmetric random ...
1
vote
0answers
54 views

Reflection principle for simple random walk

Let $(X_n)$ be a sequence of independent random variables, such that $P(X_i=1) = P(X_i=-1) = 1/2$. Then, the reflection principle states that for all $a > 0$, $$P(\max_{1\leq k\leq n} S_k \geq a) ...
1
vote
0answers
52 views

Expected value of sum of heights of books in a shelf with limited width

This question has arisen from a previous post: Statistical problem: how many books of different widths fit it into a self of a limited certain width? Let's assume that there are $N$ types of books, ...
11
votes
1answer
455 views

Does this modified random walk (2D) return with probability 1?

Pólya showed that a random walk (with the directions at each step uniformly distributed) on the integer lattice returns with probability 1. What if instead we consider the random walk where we are ...
8
votes
1answer
204 views

First player to win k matches

A series of matches are held between n identical competitors. Each is won by one of the n with equal probability (no ties). I'm looking for a probabilistic description of the outcome when looking at ...
6
votes
2answers
1k views

A question on calculating probabilities for the random walk

I am currently working on a high school project revolving around the 'Cliff Hanger Problem' taken from ”Fifty Challenging Problems in Probability with Solutions” by Frederick Mosteller. The problem ...
5
votes
1answer
319 views

Asymptotic behavior for the return to zero of a simple random walk

I got stuck today trying to understand an argument of the Frank den Hollander Book's. The problem is described below. Let $S_n=\sum_{i=1}^n X_i$ be the simple random walk in $\mathbb{Z}^d$, that is ...
4
votes
1answer
2k 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 ...
4
votes
1answer
192 views

The probability of a “double supremum” of random variable

Let $X_1,X_2,X_3,\ldots$ be IID r.v. with \begin{equation} P(X_i<-1)=0 \end{equation} \begin{equation} P(X_i<0)>0 \end{equation} \begin{equation} P(X_i>0)>0. \end{equation} Define ...