Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation
Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation
数学
Ian Waudby-Smith,Martin Larsson,Aaditya Ramdas.Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation[EB/OL].(2025-09-30)[2025-10-05].https://arxiv.org/abs/2502.06188.点此复制
We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Komlós, Major, and Tusnády (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly.
展开英文信息
评论