Random Walks and Wiener Processes

In 1827 Robert Brown noticed that small particles in a fluid were in constant motion, following irregular paths. The essential properties of Brownian motion 

To model the randomized motion of the Brownian particle, the concept of a random walk is typically used. A random walk is an example of a stochastic process, a collection of random variables parameterized by either a discrete or continuous index parameter [269, 314]. A random walk is a discrete stochastic process in which the state X at a given instant (defined by the index n) is related to the state X i, at step n — 1 by an offset that may be viewed as a random variable. That is, we have 

We initiate the random walk with Xq = 0. The quantity J is a random variable drawn from a certain probability distribution. In the classic model of a random walk the jump takes values 1 with equal probability. The increments of the random walk are the differences AX = X — X i = 8xi -, thus (X ) is a random process with independent increments. 

Expanding the squared sum we see that there are (i) n squares J plus (ii) products of the form k 1. Clearly the former are always one and the latter have zero expectation, thus 

The variance therefore grows linearly with the number of steps taken. 

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