Stochastic Processes

Any collection of random vectors {X_t,t∈T} is called a stochastic process. All the values that the random variables X_t can take on are called the state space. Because X_t is a regular random variable (or vector) we again differentiate between continuous and discrete state space cases. Moreover T can be discrete (1,2,..) or continuous (t>0) as well.

Example

Let X_i~U[0,1], i=1,2,.. X_i independent of X_j, then {X_i, i=1,2,..} is a continuous state space discrete time process.

Example

Let X_i∈{0,..,39} the position on the board of your token after i roles of the dice in a game of Monopoly. Then {X_i, i=1,2,..} is a discrete state space discrete time process.

Example

Let P(Z_i=-1)=p=1-P(Z_i=1), Z_i independent from Z_j if i≠j. Let

then X_n∈{0,±1,±2,..} and so {X_n, n=1,2,..} is a discrete state space discrete time process. This is a very famous stochastic process called a random walk.

Here is a list of things we often want to know about a stochastic process:

• what is the distribution of X_n, especially in the limit?

• what is EX_n, especially in the limit?

• what is cor(X_n,X_n+k)?