Locally stationary random processes

By R. A. Silverman. 2018

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From Stationarity to Self-similarity, and Back: Variations on the Lamperti Transformation (2003, background): “This corresponds to a class of nonstationary processes referred to as locally stationary [31], since their symmetrized covariance function is given by RY (t + τ/2, t − τ/2) = mY (t) γY (τ), (93)”
A test for second‐order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series (2013, methodology): “Probably, the most common technique for investigating this suspicion is via the regular time series plot, closely followed by applying the regular autocovariance and spectral estimators to different portions of the series to see whether they differ.”
Classes of Kernels for Machine Learning: A Statistics Perspective (2002, unknown): “In - deed , it can be written as ( Silverman , 1957 ) : K ( x , z ) = R d R d cos ω T 1 x − ω T 2 z f 1 ω 1 + ω 2 2 f 2 ( ω 1 − ω 2 ) dω 1 dω 2 , Genton i . e . the spectral density f 1 ω 1 +ω 2 2 f 2 ( ω 1 − ω 2 ) is also a locally stationary kernel , and : K 1 ( u ) = R d cos ( ω T u ) f 2 ( ω )…”
Stationary covariances associated with exponentially convex functions (2003, background): “These are required when locally stationary covariance functions in the sense of Silverman (1957) are fitted, a situation discussed by Genton and Perrin (2002).”