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1052 CSÁKI, E., CSÖRGÖ, M., FÖLDES, A. and RÉVÉSZ, P., Strong approximation of additive functionals, J. Theoret. Probab. 5 (1992), 679-706. MR 93k :60073 Strong

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CSÖRGő, M. and RÉVÉSZ, P., Strong approximations in probability and statistics , Akadémiai Kiadó, Budapest; Academic Press, New York, 1981. MR 84d :60050 Strong approximations in probability and statistics

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Abstract  

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.

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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.

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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.

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Abstract  

Consider the set Θn 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 Θn 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 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 d n 1/4(log n)−1 and are faster than in the Brownian case whereas the slowest rates are of order d n −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 α n.

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Balan, R. M. and Zamfirescu, I. M. , Strong Approximation for Mixing Sequences with Infinite Variance, Electr. Comm. Probab. , 11 (2006), 11–23. MR 2006m :60034

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