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.
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
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.
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/