Hölderian invariance principle for Hilbertian linear processes
Résumé
Let be the polygonal partial sums processes built on the linear processes , , where are i.i.d., centered random elements in some separable Hilbert space and the 's are bounded linear operators , with . We investigate functional central limit theorem for in the Hölder spaces of functions such that uniformly in , where , with and slowly varying at infinity. We obtain the weak convergence of to some valued Brownian motion under the optimal assumption that for any , when tends to infinity, subject to some mild restriction on in the boundary case . Our result holds in particular with the weight functions , .
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