Contents
The topic of martingales is both a subject of interest in its own right and also a tool that provides additional insight Rdensage into random walks, laws of large numbers, and other basic topics in probability and stochastic processes. 7.1.1 Simple random walks SupposeX 1,X
How to prepare for a symmetric random walk?
The expectation and variance of the simple symmetric random walk \\ ( W^ { (1)} (n) \\) are easy calculations that show off the power of the linearity of expectation and the delight of independent random variables. Remember that each step is of size one. Since this walk is symmetric and starts at zero, notice that \\ ( E (X_i) = 0 \\) for all \\ ( i \\).
How to calculate the probability of a random walk?
For any givenn,S nis simply a sum of IID random variables, but here the behavior of the entire random walkprocess,{S n;n ≥1}, is of interest. Thus, for a given real number α >0, we might want to find the probability that the sequence{S n;n ≥1}contains any term for whichS
What is the name of a symmetric walk?
And we know that this financial info inhabits a netherworld between discrete and continuous A simple symmetric walk has a name that makes sense. Consider a walk along a line of integers. You start at zero and move right or left one integer unit with equal probability.
Which is an example of a random walk?
The first two, simple random walks and integer random walks, will be useful throughout as examples, since they can be easily visualized and analyzed. The third special case is that of renewal processes, which we have already studied and which will provide additional insight into the general study of random walks.
How is the Bernoulli process different from a random walk?
For the Bernoulli process,X takes on the values 1 and 0, whereas for a simple random walkXtakes on the values 1 and -1. For the random walk, ifX m= 1 formout ofntrials, thenS n= 2m − n, and