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

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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" ...
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1answer
5k 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 ...
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2answers
1k 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 ...
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3answers
604 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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3answers
3k 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. ...
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1answer
158 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} ...
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1answer
371 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: ...
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1answer
580 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 ...
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1answer
196 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 ...
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1answer
291 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 ...
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1answer
522 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?
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1answer
392 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 ...
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1answer
1k views

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

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 ...
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1answer
79 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.
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2answers
300 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 ...
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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: $$ ...
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3answers
310 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 $i+1$ ...
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2answers
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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 ...
8
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6answers
346 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 ...
3
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1answer
120 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 ...
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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 ...
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1answer
155 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
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1answer
111 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
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1answer
343 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? ...
2
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1answer
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 ...
2
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2answers
187 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 = ...
2
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3answers
133 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 ...
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2answers
1k 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 ...
11
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1answer
402 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
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1answer
191 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 ...
5
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1answer
282 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
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1answer
1k 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
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1answer
191 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 ...
4
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1answer
339 views

Circular random walk

Suppose we have a circumference divided in N arcs of the same length. A particle can move on the circumference jumping from an arc to the adjacent, with probability $P_{k \to k-1}=P_{k\to ...
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1answer
787 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
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2answers
184 views

Mixing time for a random walk on an interval

I take a random walk on a bounded one-dimensional interval of length $N$, with possible walker positions $1$ through $N$. If I start at position $1$, how many steps should it take before my walker is ...
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2answers
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Definition: transient random walk

What exactly does a "transient random walk on a graph/binary tree" mean? Does it mean that we never return to the origin (assuming there is one as for the tree) or just any vertex of the graph or ...
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1answer
76 views

Visits from a transient random walker on the integers

Consider a random walk $\{S_n\}$ on $\mathbb{Z}$ with forward probability $p>\frac12$. It is known for such a transient RW that each site is a.s. visited only finitely many times. However, is it ...
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1answer
111 views

Martingales of random walk

Let $S_n$ be a random walk process defined by $$S_n=X_1+\dots+X_n$$ with $X_i \sim N(\mu,\sigma^2)$ and $X_i$ are i.i.d. I'm trying to prove that the quantity $(S_n-n\mu)^2-n\sigma^2$ is a ...
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3answers
291 views

Random walk returning probability

Consider a two-dimensional random walk, but this time the probabilities are not $1/4$, but some values $p_1, p_2, p_3, p_4$ with $\sum p_i=1$. For example, from $(0,0)$, it goes to $(1,0)$ with $p_1$, ...
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0answers
105 views

random walk with possibility to freeze

Consider a Random Walk on a one-dimensional lattice. The walker starts moving at time $0$ from $x=0$. At every step, the walker moves to the right with probability $p$, to the left with probability ...
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0answers
131 views

Repeated reflection

This is a problem from Feller's book introduction to probability theory and its application, Vol 1, Chap 3 problem 3. Let $a$ and $b$ be positive, and $-b <c<a$. Prove the number of paths to ...
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44 views

Multiple Anihilating Random Walks in a Ring (cycle)

I've been trying to solve this problem for a long time. Problem Let $R$ be a cycle with $2n$ nodes and assume there are $2k$ particles performing a simple random walk in this ring (i.e., they have ...
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1answer
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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2answers
61 views

Random walk confusion

If a ransom walk is binomial (1/2 probability of going forward, 1/2 backward) why isn;t the variance a) $\sigma=(\frac{n}{4})^.5$ b) instead of $\sigma=(n)^.5$ these sources seem to give ...
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0answers
182 views

Reference Request, Random Walk [duplicate]

The expected distance from origin after a random walk of $N$ steps in a $d$ dimensional space, is close to ...
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1answer
118 views

Average end point of 1-dimensional random walk?

Is it possible to estimate the average end point of a 1-dimensional random walk of n steps where the probability of going "left" is p and going "right" is 1-p? Thanks.
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1answer
719 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 ...
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2answers
275 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, ...
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1answer
895 views

Stationary distribution for directed graph

I want to implement the algorithm of graph partitioning of sparse directed graph. In this algorithm after computing the transition matrix ,we should compute the stationary distribution of the random ...