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Factorizable convergence of random variables in Grand Lebesgue Spaces

Maria Rosaria Formica, Eugeny Ostrovsky, Leonid Sirota

Abstract

We obtain results concerning the so-called factorization for the convergence of random variables almost everywhere (almost surely or with probability one), belonging to the classical Lebesgue-Riesz spaces and we extend these results to the Grand Lebesgue Spaces. We also give exact estimates for the parameters involved and provide several examples. We also show that the obtained estimates are, in the general case, essentially non-improvable, of course up to a multiplicative constant.

Factorizable convergence of random variables in Grand Lebesgue Spaces

Abstract

We obtain results concerning the so-called factorization for the convergence of random variables almost everywhere (almost surely or with probability one), belonging to the classical Lebesgue-Riesz spaces and we extend these results to the Grand Lebesgue Spaces. We also give exact estimates for the parameters involved and provide several examples. We also show that the obtained estimates are, in the general case, essentially non-improvable, of course up to a multiplicative constant.
Paper Structure (6 sections, 6 theorems, 83 equations)

This paper contains 6 sections, 6 theorems, 83 equations.

Table of Contents

  1. Introduction.
  2. Lebesgue - Riesz norm estimations.
  3. Grand Lebesgue Spaces norm estimations.
  4. A slight generalization.
  5. Examples. Lower estimates.
  6. Concluding remarks.

Key Result

Theorem 1.1

(Regulator) Let $\{\xi_n \}, \ n =1, 2,3,\ldots$, be a sequence of numerical valued random variables converging to zero almost everywhere (a.e.). Then there exists a non-negative finite a.e. random variable $\ \zeta \ $ and a sequence of non-random real valued non-negative variables $\epsilon_n$ t

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 1.2
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Theorem 2.2
  • ...and 8 more