Consider the set Θn of all an-sized increment processes of the uniform empirical process αn on [0, 1]. We assume that an ↓ 0, nan ↑ ∞, dn = nan(log n)−1 → ∞ and nan(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 Θn because strong approximation methods become ineffective at such a small scale an. We are now able to study directly these small empirical increments and compute the exact rate of clustering of Θn 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 dn1/4(log n)−1 and are faster than in the Brownian case whereas the slowest rates are of order dn−1/2 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 αn.