Functional Chung laws for small increments of the empirical process and a lower bound in the strong invariance principle

Abstract

Consider the set Θ _n of all a _n -sized increment processes of the uniform empirical process α _n on [0, 1]. We assume that a _n ↓ 0, na _n ↑ ∞, d _n = na _n (log _n)⁻¹ → ∞ and na _n (log n)^−7/3 = O(1). In Berthet (1996, 2005) the fourth assumption was shown to be critical with respect to the pointwise rates of convergence in the functional law of Deheuvels and Mason (1992) for Θ _n because strong approximation methods become ineffective at such a small scale a _n . We are now able to study directly these small empirical increments and compute the exact rate of clustering of Θ _n to any Strassen function having Lebesgue derivative of bounded variation by making use of a sharp small deviation estimate for a Poisson process of high intensity due to Shmileva (2003a). It turns out that the best rates are of order d ^1/4 _n (log n)⁻¹ and are faster than in the Brownian case whereas the slowest rates are of order d ^−1/2 _n and correspond to the apparently crude ones obtained in Berthet (2005) by means of Gaussian small ball probabilities. These different sharp properties of the empirical and Brownian paths imply an almost sure lower bound in the strong invariance principle and provide a new insight into the famous KMT approximation of α _n .