A functional Hungarian construction for sums of independent random variables

Abstract

We develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions $f$ from a class $\mathcal{H}$, but the supremum over $f\in $ $\mathcal{H}$ is taken outside the probability. This form is a prerequisite for the Koml\'{o}s-Major-Tusn\'{a}dy inequality in the space of bounded functionals $l^{\infty }(\mathcal{H})$, but contrary to the latter it essentially preserves the classical $n^{-1/2}\log n$ approximation rate over large functional classes $\mathcal{H}$ such as the H\"{o}lder ball of smoothness $1/2$. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments.