## 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.

Author: | Heinrich von Weizsäcker |
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URN (permanent link): | urn:nbn:de:hbz:386-kluedo-7411 |

Document Type: | Preprint |

Language of publication: | English |

Year of Completion: | 1997 |

Year of Publication: | 1997 |

Publishing Institute: | Technische Universität Kaiserslautern |

Source: | Probab. Theory Relat. Fields 107, 1997, Seiten 313-324 |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 510 Mathematik |

MSC-Classification (mathematics): | 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] |

60B11 Probability theory on linear topological spaces [See also 28C20] | |

60F05 Central limit and other weak theorems | |

60G12 General second-order processes |