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

Letbe a subset ofA family of random variablesindexed byis a stochastic or random process. Whenoris a discrete time process and whenit is a continuous time process.

Whenthe process is a single event and whenis finite the process is a random vectorx-i in the ith position indicates the ith outcome or state corresponding toThe vector represents the evolution of the system as time passes. Whenorthe changes occur discretely and whenthe changes may occur at any instant.