A functional Hungarian construction for sums of independent random variables

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 Komlós-Major-Tusnády inequality in the space of bounded functionals $l^{\mathrm{\infty }}(\mathcal{H})$, but contrary to the latter it essentially preserves the classical $n^{-1\mathrm{/}2}\log n$ approximation rate over large functional classes $\mathcal{H}$ such as the Hölder ball of smoothness $1\mathrm{/}2$. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments.