CSÖRGő, M. and RÉVÉSZ, P., Strongapproximations in probability and statistics , Akadémiai Kiadó, Budapest; Academic Press, New York, 1981. MR 84d :60050
Strongapproximations in probability and statistics
We describe the functions from Nikol’skii class in terms of behavior of their Fourier coefficients. Results for series with
general monotone coefficients are presented. The problem of strong approximation of Fourier series is also studied.
Summary Some topics of our twenty some years of joint work is discussed. Just to name a few; joint behavior of the maximum of the Wiener process and its location, global and local almost sure limit theorems, strong approximation of the planar local time difference, a general Strassen type theorem, maximal local time on subsets.
We verify a newer version of a certain embedding theorem pertaining to the relation being between strong approximation and
a certain wide class of continuous functions. We also show that a new class of numerical sequences defined in this paper is
not comparable to the class defined by Lee and Zhou, which is one of the largest among the classes being extensions of the
class of monotone sequences.
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.