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Let ${\left({\xi }_n\right)}_{n\ge 1}$ be the polygonal partial sums processes built on the linear processes $X_n={\sum }_{i\ge 0}{a}_i\left({ϵ}_{n-i}\right)$, ≥ 1, where ${\left({ϵ}_i\right)}_{i\in \mathbb{Z}}$ are i.i.d., centered random elements in some separable Hilbert space $ℍ$ and the 's are bounded linear operators $ℍ\to ℍ$, with ${\sum }_{i\ge 0}\parallel a_i\parallel <\infty$. We investigate functional central limit theorem for ${\xi }_n$ in the Hölder spaces ${\mathrm{H}}_{\rho }^o\left(ℍ\right)$ of functions $x:[0,1]\to ℍ$ such that || uniformly in , where , 0 ≤ ≤ 1 with 0 ≤ ≤ 1/2 and slowly varying at infinity. We obtain the ${\mathrm{H}}_{\rho }^o\left(ℍ\right)$ weak convergence of ${\xi }_n$ to some $ℍ$ valued Brownian motion...

We consider stochastic processes as random elements in some spaces of Hölder functions vanishing at infinity. The corresponding scale of spaces $C_0^{\alpha ,0}$ is shown to be isomorphic to some scale of Banach sequence spaces. This enables us to obtain some tightness criterion in these spaces. As an application, we prove the weak Hölder convergence of the convolution-smoothed empirical process of an i.i.d. sample $(X_1,...,X_n)$ under a natural assumption about the regularity of the marginal distribution function F of the...

Invariance principle in $L^2(0,1)$ is studied using signed random measures. This approach to the problem uses an explicit isometry between $L^2(0,1)$ and a reproducing kernel Hilbert space giving a very convenient setting for the study of compactness and convergence of the sequence of Donsker functions. As an application, we prove a $L^2(0,1)$ version of the invariance principle in the case of $\varphi$-mixing random variables. Our result is not available in the $D(0,1)$-setting.