Given a stationary continuous-time process f (t), the Hilbert-Schmidt operator Aτ can be defined for every finite τ [3]. Let λτ,i be the eigenvalues of Aτ with descending order. In this article, a Hilbert space H f and the (time-shift) continuous one-parameter semigroup of isometries K s are defined. Let {vi, i ∈ N} be the eigenvectors of K s for all s ≥ 0. Let f = ∞ i=1 aivi + f ⊥ be the orthogonal decomposition with descending |ai|. We prove that lim τ →∞ λτ,i = |ai| 2 . The continuous oneparameter semigroup {K s : s ≥ 0} is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on L 2 (X, ν), if the dynamical system is ergodic and has invariant measure ν on the phase space X.
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