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Necessary and Sufficient Conditions for the Lacunary/Hereditary Laws of Large Numbers

Istvan Berkes, Ioannis Karatzas, Walter Schachermayer

Abstract

The celebrated theorem of Komlos asserts that L1-boundedness is sufficient for a given sequence of functions to contain a subsequence along which (in a "lacunary" manner), and along whose every further subsequence ("hereditarily"), a strong law of large numbers holds. We identify here slightly weaker, Egorov-type conditions, as not only sufficient in this context, but necessary as well. Necessary and sufficient conditions are developed also for the lacunary/hereditary version of the weak law of large numbers for general sequences, as well as for the weak law of large numbers in the context of exchangeable sequences, both long-open questions.

Necessary and Sufficient Conditions for the Lacunary/Hereditary Laws of Large Numbers

Abstract

The celebrated theorem of Komlos asserts that L1-boundedness is sufficient for a given sequence of functions to contain a subsequence along which (in a "lacunary" manner), and along whose every further subsequence ("hereditarily"), a strong law of large numbers holds. We identify here slightly weaker, Egorov-type conditions, as not only sufficient in this context, but necessary as well. Necessary and sufficient conditions are developed also for the lacunary/hereditary version of the weak law of large numbers for general sequences, as well as for the weak law of large numbers in the context of exchangeable sequences, both long-open questions.
Paper Structure (30 sections, 19 theorems, 112 equations)

This paper contains 30 sections, 19 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Sufficiency and Necessity in the Lacunary/Hereditary SLLN
  3. Stable Convergence
  4. Integrability of the Limit in Theorem \ref{['Thm2.1']}
  5. Sufficiency and Necessity in the Lacunary/Hereditary WLLN
  6. General Sequences
  7. The WLLN for Exchangeable Sequences
  8. The Classical Case
  9. The Exchangeable Case
  10. Some Counterexamples
  11. Consequences in the General Setting
  12. The Proof of Theorem \ref{['Thm2.1']}
  13. Proof of Theorem \ref{['Thm2.1']} (i)
  14. Proof of Theorem \ref{['Thm2.1']} (ii)
  15. Proof of Theorem \ref{['Thm2.1']} (iii)
  16. ...and 15 more sections

Key Result

Theorem 2.1

Sufficiency/Necessity in the Lacunary/Hereditary SLLN for General Sequences: On a probability space $\,(\Omega, \mathcal{F}, {\mathbb P}),$ consider real-valued, measurable functions $f_1, f_2, \cdots$. (i) Suppose that for some subsequence $\,f_{k_1}, f_{k_2}, \cdots\,$ and sets $\, A_1 \subseteq

Theorems & Definitions (23)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • Theorem 3.1
  • Remark 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Proposition 4.3
  • Corollary 4.4
  • Lemma 5.1
  • ...and 13 more