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)

References

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