# The Random Walk

Here is a short guide to the random walk found in probability. Note that this random walk will be a symmetrical random walk with equal probabilities of for each of the two outcomes.

Introduction

Let us first consider a unbiased/even coin. We have a probability of 0.5 for heads and 0.5 for tails.

Denote the outcome of tosses as () is omega) for coin tosses. is an infinite sequence of outcomes up to .

Define the random variable  for the coin toss for from 1 to n where:

Each coin toss is either +1 for a heads or a -1 for a tails with a probability of 0.5 each.

Note that this random variable does not have to relate to coin tosses. Once can define to be dependent on up/down movements, even/odd numbers, etc.

Mean and Variance of

since .

since

The Symmetrical Random Walk

Now we have the random variable for j from 1 to n. But, what if we want a running total of these +1 and -1 outcomes for 1 to n?

Let us define and this “running total” as where:

for k = 1, 2, …
This stochastic (random) process is a symmetric random walk.

Properties of the Symmetric Random Walk

1) Mean is zero:

.

2) Variance is just for .

.

Note that the independence of coin tosses was assumed such that the covariance in the double sum is zero.

3) The symmetric random walk is a martingale. That is . (The conditional expectation given the filtration at time is just the symmetric random walk at time k.) We don’t expect the symmetric random walk to change from time to .

4) The quadratic variation of the symmetric random walk is just time k. This is because: