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- CONTENTS 1987 - VOLUME 305 - SECTION I - N° 19
- .......... Page(s) .......... 801
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- We give a general result concerning the homogenization of scattering cross sections in Transport theory, still valid far from the diffusion regime, by using the compactness of the moments of the solution of a Transport equation proved in[1].We also study the homogenization of opacities in Radiative Transfer, on some simplified models.
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- It is shown under reasonable assumptions that a given vector fielddefined on a compact setKofRdcan be factored in the form,where u is a convex function defined onKand g is a volume preserving mapping fromKinto itself. This factorization, closely linked to the Monge-Ampère equation, generalizes the polar decomposition of matrices, as well as the increasing rearrangement of real functions, and the De Rham decomposition of vector fields.
- .......... Page(s) .......... 845
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- For a holomorphic function with isolated singularity at0,f:(Cn, 0)(C, 0)we prove that for any morsification, there exists a system of paths such that the corresponding vanishing cycles are unknotted in the Milnor sphereS().For n3this result follows easily from Whitney embedding theorems. So the real problem is in dimension two.
- .......... Page(s) .......... 827
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- We first extend the Orlicz inequality for covariances given by Bulinskii(1987)to the Hilbert valued case. After this we show generalizations of Marcinkiewicz-Zygmund inequality of higher order moments for sums of strongly mixing random variables extending to Orlicz case results by Doukhan-Portal(1983)and Doukhan-Léon-Portal(1984).Orlicz norms allow weakening of moment assumptions. Interest of the results is explicited by the example of kernel density estimates.
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- We show the existence of a limit law for a sequence of uniform Lipschitz random iterates. The law is investigated and the results are applied to obtain the asymptotic behaviour of the state estimation error in a Kalman filtering operated along a renewal process.
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- Letsbe any increasing sequence of integers andM>1;we associate to them in a simple way, an increasing unbounded map.Let alsoX1, X2...be a sequence of i. i. d. random vectors with value in euclidean spaceRm.We prove that the cluster set of the sequencealmost surely coincides with the unit ball ofRm,if and only if, the covariance matrix ofX1,is the identity matrix ofRmandEX1is the zero vector ofRm.We define a functionalAon the set of strictly increasing sequences of integers as follows:
- We prove that,for at least one sequenceX1, X2, ...of
- i. i. d. real random variables withEX1= 0andE(X1)2= 1,if and only if;further the definition ofA(.)does not depend on the value ofM.Further, the law of the iterated logarithm for subsequences in the sense of Strassen is considered. We finally show a functional law of the iterated logarithm on subsequences for lipschitzian random functions.
- .......... Page(s) .......... 841
- We study the convergence in law of normalized additive functionals of a Markov process (in discrete or continuous time) which is recurrent in Harris sense. Our results are obtained under a weak ergodic hypothesis on the behaviour near0of the resolvant of this processes.
- MATHEMATIQUE 1987 - Tome 305 - Série I - n° 19
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