# 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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### Reflection principle for walk possible steps right, left and stay

I need to use reflection principle for one dimensional walk with equaly possible steps right, left and stay. I would like to know if hold a similiar identity to that of question Is there an intuitive ...
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### Formula to explain table data.

This is part of an excel spreadsheet and I would like to use a formula instead of a table to calculate steps and probabilities and so forth. It is my understanding that the table below represents the ...
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### Probability generating function of some “random walk”

Let $S_n=\sum^n_{r=0}X_r$ be a left-continuous random walk on the integers with a retaining barrier at zero. More specifically, we assume that the $X_r$ are identically distributed integer-valued ...
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### Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...
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### Is Markov Chain property true for correlated inputs?

I have a finite state machine (FSM). At time $k$, state is $\theta^k$ and input is $x^k$. The next state $\theta^{k+1}$ and output $y^k$ are completely determined by \begin{align} \theta^{k+1} &=...
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### Confusion about the average distance traveled on a $1$D random walk

The average absolute distance on a one dimensional random walk is supposed to be $\sqrt{n}$. Where $n$ steps are taken from the origin or $n$ is the time. I don't have an intuitive understanding or ...
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### 1D random walk probability distribution

I am way more physicist than mathematician and this question arises from experimental physics/engineering. The motivation is dealing with small amount of random discrete shifts between measured ...
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### The probability that the d-dimensional symetric random walk returns to the origin - is this relatively short proof correct?

Let $p_n$ denote the probability of returning to the origin after n steps. If n is odd, $p_n = 0$. The main insight is that $\sum_{n=0}^{\infty}p_{2n}$ is asymptotically ~ $C \cdot \frac{1}{n^{d/2}}$ ...
Consider a random walk $(X_n)$ on the graph below, where we jump from a given vertex to one of its adjacent vertices with equal probability. I want to find: the probability that we hit $A$ before ...