Stochastic Processes ( Fall 2014 ) Spectral representations and ergodic theorems for stationary stochastic processes Stationary stochastic processes

A stochastic process X is strongly stationary if its fdds are invariant under time shifts, that is, for any (finite) n, for any t0 and for all t1, ..., tn ∈ T , (Xt1 , ..., Xtn) and (Xt1+t0 , ..., Xtn+t0) have the same distribution. A stochastic process X is weakly stationary if its mean function is constant and its covariance function is invariant under time shifts. That is, for all t ∈ T , E(Xt) = μ, and for all ti, tj ∈ T , Cov(Xti , Xtj ) = c(ti − tj), a function of ti − tj only. (Note that the definition of weak stationarity implicitly assumes existence of first and second order moments of the process.)