Upper-Lower Class Tests for Weighted i.i.d. Sequences and Martingales

We study the upper-lower class behavior of weighted sums ∑ _k=1 ⁿ a _k X _k , where X _k are i.i.d. random variables with mean 0 and variance 1. In contrast to Feller’s classical results in the case of bounded X _j , we show that the refined LIL behavior of such sums depends not on the growth properties of (a _n ) but on its arithmetical distribution, permitting pathological behavior even for bounded (a _n ). We prove analogous results for weighted sums of stationary martingale difference sequences. These are new even in the unweighted case and complement the sharp results of Einmahl and Mason obtained in the bounded case. Finally, we prove a general upper-lower class test for unbounded martingales, improving several earlier results in the literature.     

Optimization of some instrumental factors in diffuse reflectance infrared Fourier transform spectroscopy

The effects of some instrumental factors on infrared spectroscopy analysis were investigated in the case of diffuse reflectance infrared Fourier transform (DRIFT) mode. It is usually said that quantitative analysis is possible only if both particle size distribution and sample density are perfectly controlled. However, even if these conditions are checked, instrumental factors are of great interest for the goodness of a curve fitting procedure, which is often necessary in solid sample studies. A factorial design achieved on anthron made it possible to obtain major trends concerning the required values for one instrumental parameter (resolution) and two mathematical treatments (zero filling and Savitsky-Golay (S-G) smoothing). Resolution was found to have the greatest effect on measured responses. A value of 2 cm(-1) according to the corresponding aperture is sufficient to approach the real width of bands for powdered samples. The use of a zero filling factor (ZFF) improves the apparent resolution by data interpolation. The best values found for instrumental parameters were applied to an anthron-anthracen mixture. The optimization of spectral features made it possible to obtain semi-quantitative results quite easily, in good agreement with the data corresponding to each pure compound.     

Régularisation spectrale et propriétés métriques des moyennes mobiles

We develop a spectral regularization technique for moving averages , where ϕ is a nondecreasing map andU: H→H is a contraction of a Hilbert space (H, ‖·‖). We obtain a spectral regularization inequality which allows one to evaluate efficiently the increments ‖B _m ^U , ϕ (f)−B _n ^U , ϕ (f)‖,f∈H, by means of where is a properly regularized version of the spectral measure off with respect toU. We apply this inequality to an investigation of metric properties of the sets of moving averages {B _n ^{U, ϕ} (f), n ∈N} with fixedf∈H andN⊂ N. In particular, we obtain estimates of the associated covering numbers as well as of the related Littlewood-Paley-type square functions. This work extends our previous results concerning the case of classical averages (ϕ(n)=0). Since it is well-known that the structure of general moving averages is more complicated, there is no surprise that the general results we obtain are sometimes less complete than their classical counterparts and need suitable moment assumptions on the spectral measure (depending on the growth of the shift function ϕ). Nevertheless, when applied to the classical situation, our estimates still yield optimal bounds.

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Avec pour le premier author, le soutien de la fondation russe pour la recherche fondamentale, subvention 99-01-00112 et INTAS subvention 99-01317.