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

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  • Chambers, Daniel
  • Slud, Eric

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.

Suggested Citation

  • Chambers, Daniel & Slud, Eric, 1989. "Necessary conditions for nonlinear functionals of Gaussian processes to satisfy central limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 32(1), pages 93-107, June.
    • Handle: RePEc:eee:spapps:v:32:y:1989:i:1:p:93-107
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    Cited by:

    1. Serge Cohen & Mario Wschebor, 2010. "On Tightness and Weak Convergence in the Approximation of the Occupation Measure of Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1204-1226, December.

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