A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Ruin-time in Gambler's Ruins

Could someone give me some tips? Let $e_1,e_2,\dots$ be iid normal mean 0 variance 1. Let $X_t := e_1+\cdots+e_t$, for $t=1,2,\dots$ and $X_0 := 0$. (So we have a discrete-time random walk whose ...
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389 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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Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$

Some years ago I was interested in the following Markov chain whose state space is the positive integers. The chain begins at state "1", and from state "n" the chain next jumps to a state uniformly ...
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2answers
182 views

Is there an analytic solution for the density function of this complex random variable?

The process below yields a distribution of "response times" (RT), and I'd like to know if there is an analytic solution to obtain the density function of this distribution. An RT is recorded at the ...
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5answers
232 views

Explanation of numeric experiment that approximates e?

Recently I found this post on Reddit. It describes the following algorithm to find e: Here is an example of e turning up unexpectedly. Select a random number between 0 and 1. Now select ...
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336 views

Analysis of a biased random walk related to median finding

Imagine a process with two variables min and max, and two counters hi and lo. We initialize min and max by selecting two random numbers (assume a uniform (0,1) distribution for convenience), and ...
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537 views

Taylor expansion to show that for Stratonovich stochastic calculus the chain rule takes the form of the classical one

How can I show with a heuristic argument based on a Taylor expansion that for Stratonovich stochastic calculus the chain rule takes the form of the classical (Newtonian) one? Concerning Ito calculus ...
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What is a Markov Chain?

What is a intuitive explanation of a Markov Chain, and how they work? Please provide at least one practical example.
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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 ...