Learning Notes

# Stochastic – Stationary Process Stochastic

## Definition

### Strictly Stationary Process

A stochastic process $X_T = \{X_t, t\in T\} = \{X(t), t\in T\}$, with T being a totally ordered set (which usually denotes time), is strictly stationary process (SSS) if its mapping is invariant under time. i.e. For its n-dimensional outcome:

$\displaystyle \{X(t_1), X(t_2), \dots, X(t_n)\} = \{X(t_1 + h), \dots, X(t_n + h)\}$

where $\{t_i, t_i + h \in T ; i \in \{1, 2, \dots, n\}\}$

### Weakly Stationary Process

A stochastic process $X_T$ is weakly stationary or covariance stationary if its mean function or expectation value remains constant against variation of T. Expectation value can be defined formally:

$\displaystyle \text{exp}(X(t)) = \int_{T} X(t)dP(t) = \int_{-\infty}^{\infty} x dP_X (x)$