The functional Breuer-Major theorem

18 May 2020
Ivan Nourdin

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Let ๐‘‹={๐‘‹๐‘›}๐‘›โˆˆโ„ค be zero-mean stationary Gaussian sequence of random variables with covariance function ฯ satisfying ฯ(0)=1. Let ฯ†:Rโ†’R be a function such that ๐”ผ[๐œ‘(๐‘‹_0)2]<โˆž and assume that ฯ† has Hermite rank dโ‰ฅ1. The celebrated Breuerโ€“Major theorem asserts that, if โˆ‘|๐œŒ(๐‘Ÿ)|^๐‘‘<โˆž then
the finite dimensional distributions of the normalized sum of ๐œ‘(๐‘‹๐‘–) converge to those of ๐œŽ๐‘Š where W is
a standard Brownian motion and ฯƒ is some (explicit) constant. Surprisingly, and despite the fact this theorem has become over the years a prominent tool in a bunch of different areas, a necessary and sufficient condition implying the weak convergence in the
space ๐ƒ([0,1]) of cร dlร g functions endowed with the Skorohod topology is still missing. Our main goal in this paper is to fill this gap. More precisely, by using suitable boundedness properties satisfied by the generator of the Ornsteinโ€“Uhlenbeck semigroup,
we show that tightness holds under the sufficient (and almost necessary) natural condition that E[|ฯ†(X0)|p]<โˆž for some p>2.

Joint work with D Nualart

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