# Tagged Questions

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

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### Mean absolute distance for a symmetric random walk

I have found that the mean absolute distance for a symmetric random walk after n steps can be computed using this product: What can be deduced from this? For example how can variance be computed ...
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### The boundedness of a certain sequence of expectations

In Bálint Tóth's paper, "No More Than Three Favourite Sites for Simple Random Walk", while proving one of the many technical lemmas in his theorem's proof, he makes the following claim: suppose for ...
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### Help with a random walk problem.

Let $\xi_1,\xi_2,...$ be a sequence of random variables such that $\xi_i=-1$ or $\xi_i=1$ for $i=1,2,...$. Let $x(n)$ be the position of a random walk at time $n$, i.e. $x(n)=\xi_1+\xi_2+...+\xi_n$. ...
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### Why is this true with random walks?

Let $S_n=\sum_n{X_n}$ a random walk with $P[X=1]=p=1-P[X=-1]$. Prove that for any $k \in Z$ $P[\cup_{n \geq1} \{S_n\geq k\}]= (P[\cup_{n \geq1} \{S_n\geq 1\}])^k$ I do not understand why is this ...
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### Calculating infinite sum from a random walk in 1-dimension

Consider the random walk on $\mathbb{Z}$ with 1-step transition probabilities $p_{i,j} = \frac{1}{2}$ iff $|i - j| = 1$. Then the probability that the first time that the chain returns from $i$ to $i$ ...
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### How does one reconcile the fact that a random walk is non-mean reverting with Polyas recurrence theorem?

In particular, Polyas theorem says that in 2 or 1 dimension the symmetric random walk is recurrent so that it has probability 1 of returning to zero. Yet, I have read that random walks are known as ...
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### Random walk in $1$ dimension with non-equal left and right probabilty

Consider a typical random walk problem, where the probability to go right is $R$ and the probability to go left is $L$, where $R+L=1$. The particle can move 1 unit in each step, and starts at zero. ...
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### probabilistically segmenting a rectangle

I am trying to find ways to segment a image randomly, but drawn from a probalistic distribution of pre-determined areas to be cut through. First thought was to pick random points and run Covex hull ...
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### random walk on rotation matrix

I have a $3\times3$ matrix and I have to create a random process that rotates this matrix and such that there is a typical time of decorrelation of the matrix: the mean time needed to reorient the ...
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### Most likely position for random walk with symmetric jumps after $t$ steps

Consider a random walk on $\mathbb{Z}$ starting at 0 with jump distribution $p(x)$ such that, $p(x) = p(-x)$ $p(x)>0$ for all $x \in \mathbb{Z}$ Let $p^{\,t}(n)$ be the probability that the ...
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### Maximum of random walk distribution for arbitrary jump distribution and even times

Consider a random walk with an arbitrary jump distribution on $\mathbb{Z}$ and zero expected increment. Let $p^t_n$ be the probability that at time $t$ the random walk is at $n \in \mathbb{Z}$. Is it ...
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### Infinite expected number of visits with Green function

The random walk Green function is defined by $G(x,1) = \sum_{n\in \mathbb{N}_0} P(S_n = x)$, with $x\in\mathbb{Z}^d$. $G(x,1)$ equals the expected number of visits to $x$. Now I want to prove that ...
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### 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 \...
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### Clarification on Two-player Gambler's Ruin variant.

I looked around and found many questions related to two-player variants of Gambler's Ruin but could not find what I had in mind. I needed clarification on a certain step while deriving the expression. ...
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### Hitting probabilities for random walk with +m/-n steps

Random walk theory in the real axis: a frog starts at k initially, and in each move, it moves either by distance m to the right (from i to i+m) or by n to the left(from i to i-n), where k,m,n are ...
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### Expected distance of a random walk of distance $k$ on the $k$th step

I am trying to sharpen my intuition on some random-walk style results. Suppose we are looking at a random walk on $\mathbb{Z}$ starting at $0$. At the $k$th step, we either walk to the left $k$ ...
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### 2-dimensional random walk experiment

Question: Consider a board covered by square tiles(like a chess set only of a size of your choosing) and colored like a chess set. Two "people" are placed in different areas of the board and ...
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Let $S_n=X_1+...+X_n$, $n\geq1$ be a random walk, where $EX_k=\mu$ and $Var(X_k)=\sigma^2$, $0<\sigma^2<\infty$. a)Find the covariance $Cov(S_n,S_m)$ and the correlation coefficient $\rho(S_n,... 1answer 243 views ### Last vertex visited by the symmetric random walk on a discrete circle$n$cats form a circle, indexed from$0$to$n-1$. At first, there is a ball at the cat$0$. We throw a coin with the probability of$p$heads up. If the coin is heads up, we pass the ball clockwise, ... 0answers 30 views ### Random walks with limited number of visits I'm interested in random walks (esp. their hitting times) such that the number of visits to each state is limited by some parameter$K$. Is there any canonical name for such stochastic processes? ... 0answers 86 views ### Random walks intuition [closed] I came across the reflection principle and this explanation at Is there an intuitive way to see this property of random walks? However, I can't understand the reasoning in the answer about mirroring ... 1answer 20 views ### Formula for the analytical computation of Eigenvalues for random walks of order one and two on regular lattices I have a question with respect to random walk penalties in statistics. In Bayesian statistics an adequate prior for modeling a$n$-dimensional random vector$\boldsymbol{x}$as a random walk of order$...
I am trying to calculate the first two moments of the random walk below: $y_t$=$y_0$+$\sum_{j=1} ^t u_j$ Mean: E[$y_t$]=$y_0$ Variance: E[$y_t ^2$]=t$\sigma^2$ I understand how to obtain the 1st ...