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# Introduction to Stochastic Processes

Summary: Describes signals that cannot be precisely characterized.

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## Definitions, distributions, and stationarity

Definition 1: Stochastic Process
Given a sample space, a stochastic process is an indexed collection of random variables defined for each ωΩ ω Ω .
X t ω for all tRωΩ X t ω  for all  t ω Ω
(1)

### Example 1

For a given t, X t ω X t ω is a random variable with a distribution

### First-order distribution

F X t b=Pr X t b=Pr ωΩ X t ωb F X t b Pr X t b Pr ω Ω X t ω b
(2)

Definition 2: First-order stationary process
If F X t b F X t b is not a function of time then X t X t is called a first-order stationary process.

### Second-order distribution

t 1 , t 2 , b 1 , b 2 , t 1 R t 2 R b 1 R b 2 R: F X t 1 , X t 2 b 1 b 2 =Pr X t 1 b 1 X t 2 b 2 t 1 t 2 b 1 b 2 t 1 t 2 b 1 b 2 F X t 1 , X t 2 b 1 b 2 Pr X t 1 b 1 X t 2 b 2
(3)

### Nth-order distribution

F X t 1 , X t 2 , , X t N b 1 b 2 b N =Pr X t 1 b 1 X t N b N F X t 1 , X t 2 , , X t N b 1 b 2 b N Pr X t 1 b 1 X t N b N
(4)

Definition 3: Nth-order stationary
A random process is stationary of order N if
F X t 1 , X t 2 , , X t N b 1 b 2 b N = F X t 1 + T , X t 2 + T , , X t N + T b 1 b 2 b N F X t 1 , X t 2 , , X t N b 1 b 2 b N F X t 1 + T , X t 2 + T , , X t N + T b 1 b 2 b N
(5)
Definition 4: Strictly stationary
A process is strictly stationary if it is Nth order stationary for all N.

For examples of stochastic processes, please refer to Examples of Stochastic Processes.

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