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Ergodic properties of locally stationary processes. (English) Zbl 0612.60035

Let x(t) be a locally stationary process with \(Ex(t)=0\) and the covariance function R(s,t) satisfying the condition: \(R(s,t)=R_ 1(s+t)R_ 2(s-t)\). It is assumed that x(t) is harmonizable, i.e., can be expressed as a stochastic integral: \[ x(t)=\int^{+\infty}_{- \infty}\exp (it\lambda)dX(\lambda) \] where X(\(\lambda)\) is a second order stochastic process with a bounded covariance function. The paper starts with a lemma demonstrating that stochastic integrals of x(t) can be expressed as integrals of X(\(\lambda)\): \[ \int^{T}_{- T}x(t)dt=\int^{+\infty}_{-\infty}\{\int^{T}_{-T}\exp (it\lambda)dt\}dX(\lambda). \] Three ergodic theorems follow, for various sets of assumptions on the covariance function R(s,t).
Reviewer: L.Podkaminer

MSC:

60G10 Stationary stochastic processes
60H05 Stochastic integrals
60F20 Zero-one laws
60E10 Characteristic functions; other transforms
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References:

[1] J. Michálek: Spectral decomposition of locally stationary random processes. Kybernetika 22 (1986), 3, 244-255. · Zbl 0601.60036
[2] M. Loève: Probability Theory. D. van Nostrand, Toronto-New York-London 1955. · Zbl 0066.10903
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