their equilibrium steady-state, but continuously undergoing random'func tions. This is the class of stationary stochastic processes Definition A stochastic process Xt, tET is said to be(strictly) stationary if any subset (t1, t2, ...,tr) of T and any T, 1+r;…tr+ That is, the distribution of the process remains unchanged when shifted in time by an arbitrary value T. In terms of the marginal distributions, (strictly)stationarity implies that F(X)=F(Xt+),t∈T, and hence F(t= F(t,)= F(rtr). That is stationarity implies that Xt, Xt, Xtr are(individually) identically distributed The concept of stationarity, although very useful in the context of probability theory, is very difficult to verify in practice because it is defined in terms of dis- tribution function. For this reason the concept of the second order stationarity defined in terms of the first two moments, is commonly preferred A stochastic process Xt, tET is said to be(weakly) stationary if E(Xt= u for all t; t4,t)=E(X4-)(x2-1)=m-4,t,t∈T These suggest that weakly stationarity for Xt, tE T) implies that its mean and variance u(ti)=7o are constant and free of t and its autocovariance depends on the interval t;-til; not ti and t;. Therefore, %k=)-k Example Consider the normal stochastic process in the above example. With the weaklytheir equilibrium steady − state, but continuously undergoing ’random’ func￾tions. This is the class of stationary stochastic processes. Definition: A stochastic process {Xt ,t ∈ T } is said to be (strictly) stationary if any subset (t1,t2, ...,tT ) of T and any τ , F(xt1 , ..., xtT ) = F(xt1+τ , ..., xtT +τ ). That is, the distribution of the process remains unchanged when shifted in time by an arbitrary value τ . In terms of the marginal distributions, (strictly) stationarity implies that F(Xt) = F(Xt+τ ), t ∈ T , and hence F(xt1 ) = F(xt2 ) = ... = F(xtT ). That is stationarity implies that Xt1 , Xt2 , ..., XtT are (individually) identically distributed. The concept of stationarity, although very useful in the context of probability theory, is very difficult to verify in practice because it is defined in terms of dis￾tribution function. For this reason the concept of the second order stationarity, defined in terms of the first two moments, is commonly preferred. Definition: A stochastic process {Xt ,t ∈ T } is said to be (weakly) stationary if E(Xt) = µ for all t; v(ti ,tj) = E[(Xti − µ)(Xtj − µ)] = γ|tj−ti| , ti ,tj ∈ T . These suggest that weakly stationarity for {Xt ,t ∈ T } implies that its mean and variance v 2 (ti) = γ0 are constant and free of t and its autocovariance depends on the interval |tj − ti |; not ti and tj . Therefore, γk = γ−k. Example: Consider the normal stochastic process in the above example. With the weakly 5