Given a stationary Gaussian vector process x sub m, ym an element of Z, and two real functions H(x) and K(x) we define Z sub H superscript N define Sum from m=1 to (n-1) of Inverse A sub n Sum from m=1 to (n-1) of Sub m and Sub K superscript k Inverse B Sub n Sum from m=1 to (n-1) of Sub n where An and Bn are some appropriate constants. The joint limiting distribution of Sub H superscript n Sub k superscript k is investigated. It is shown that Sub H superscript n and Sub k superscript k are asymptotically independent when one of them satisfies a central limit theorem. The application of this to the limiting distribution for a certain class of non-linear infinite-coordinated functions of a Gaussian process is also discussed. Keywords: Central limit theorem; Nin-central limit theorem; Long range dependence; Stationary Gaussian vector processes.
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