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:

**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:

(Add 1 k times).

**5)** The increments of the symmetric random walk are independent. For example,

and are independent increments. This means that increments over non-overlapping intervals are independent since the intervals depend on different coin tosses.

**Summary**

The symmetric random walk is a “running total” on the random variable . This random variable is either +1 or -1 with equal probabilities from one of two outcomes (heads / tails for example). The symmetric random walk has a lot useful properties and is useful for understanding Brownian Motion.

The featured image was taken from http://blog.motheyes.com/2010/03/ministry-of-stochastic-walks/