A Brief Introduction
Stochastic processes model sequences of outcomes such that the outcome or state at each stage depends on previous outcomes or previous states. A typical example is a random walk. At each stage a machine may turn up, down, left or right. If we consider the outcome to be the direction of the movement and the state to be the position of the machine, obviously the position at each stage depends on on the previous position.
Definition
Let
be a subset of
A family of random variables
indexed byis a stochastic or random process. When
or
is a discrete time process and when
it is a continuous time process.
When
the process is a single event and when
is finite the process is a random vector
x-i in the ith position indicates the ith outcome or state corresponding to
The vector represents the evolution of the system as time passes. Whenor
the changes occur discretely and whenthe changes may occur at any instant.
