## Abstract

We prove extensions of Menchoff's inequality and the MenchoffRademacher theorem for sequences {f_{n}} ⊂ L_{p}, based on the size of the norms of sums of sub-blocks of the first n functions. The results are applied to the study of a.e. convergence of series ∑_{n} a _{n}T^{n}g/n^{α} when T is an L _{2}-contraction, g ε L_{2}, and {a_{n}} is an appropriate sequence. Given a sequence {f_{n}} sub; L _{P}(Ω,μ), 1 < p ≤ 2, of independent centered random variables, we study conditions for the existence of a set of x of μ-probability 1, such that for every contraction T on L _{2}(γ,π) and g ε L_{2}(π), the random power series ∑_{n} f_{n}(x)T^{n}g converges π-a.e. The conditions are used to show that for {f_{n}} centered i.i.d. with f _{1} ε L log^{+} L, there exists a set of x of full measure such that for every contraction T on L_{2}(γ, π) and g ε L_{2}(π), the random series ∑_{n} fn(x)T ^{n}p/n converges π-a.e.

Original language | English |
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Pages (from-to) | 41-86 |

Number of pages | 46 |

Journal | Israel Journal of Mathematics |

Volume | 148 |

DOIs | |

State | Published - 1 Dec 2005 |

## ASJC Scopus subject areas

- Mathematics (all)