## Sudakov's typical marginals, random linear functionals and a conditional central limit theorem

• V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a highdimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure P and a random (a.s.) linear functional F on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of P under F are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions.

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Author: Heinrich von Weizsäcker urn:nbn:de:hbz:386-kluedo-7411 Preprint English 1997 1997 Technische Universität Kaiserslautern 2000/04/03 Probab. Theory Relat. Fields 107, 1997, Seiten 313-324 Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 28-XX MEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX) / 28Cxx Set functions and measures on spaces with additional structure [See also 46G12, 58C35, 58D20] / 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Bxx Probability theory on algebraic and topological structures / 60B11 Probability theory on linear topological spaces [See also 28C20] 60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Fxx Limit theorems [See also 28Dxx, 60B12] / 60F05 Central limit and other weak theorems 60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Gxx Stochastic processes / 60G12 General second-order processes Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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