Necessary conditions for nonlinear functionals of Gaussian processes to satisfy central limit theorems

Daniel Chambers and Eric Slud

Stochastic Processes and their Applications, 1989, vol. 32, issue 1, 93-107

Abstract: Let be a stationary Gaussian process on ([Omega], , P) with time-shift operators (Us, s [epsilon] ) and let H(X) = L2([Omega], [sigma](X), P) denote the space of square-integrable functionals of X. Say that Y [epsilon] H(X) with EY = 0 satisfies the Central Limit Theorem (CLT) if A family of martingales (ZT(t), t [greater-or-equal, slanted] 0) is exhibited for which ZT([infinity]) [reverse not equivalent] ZT, and martingale techniques and results are used to provide sufficient conditions on X and Y for the CLT. These conditions are then shown to be necessary for slightly more restrictive central limit behavior of Y.

Date: 1989
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