Let ${\left({\xi }_n\right)}_{n\ge 1}$ be the polygonal partial sums processes built on the linear processes $X_n={\sum }_{i\ge 0}{a}_i\left({ϵ}_{n-i}\right)$, n ≥ 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 ai'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 ||x(t + h) - x(t)|| = o(p(h)) uniformly in t, where p(h) = hαL(1/h), 0 ≤ h ≤ 1 with 0 ≤ α ≤ 1/2 and L slowly varying at infinity. We obtain the ${\mathrm{H}}_{\rho }^o\left(ℍ\right)$ weak convergence of ${\xi }_n$...

This paper deals with homogenization of second order divergence form parabolic operators with locally stationary coefficients. Roughly speaking, locally stationary coefficients have two evolution scales: both an almost constant microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims to give a macroscopic approximation that takes into account the microscopic heterogeneities. This paper follows [Probab. Theory Related Fields (2009)] and improves this latter work by...

The asymptotic behavior of global errors of functional estimates plays a key role in hypothesis testing and confidence interval building. Whereas for pointwise errors asymptotic normality often easily follows from standard Central Limit Theorems, global errors asymptotics involve some additional techniques such as strong approximation, martingale theory and Poissonization. We review these techniques in the framework of density estimation from independent identically distributed random variables,...