Stable limit theorems for empirical processes under conditional neighborhood dependence

JiHyung Lee, Kyungchul Song

Research output: Contribution to journalArticlepeer-review

Abstract

This paper introduces a new concept of stochastic dependence among many random variables which we call conditional neighborhood dependence (CND). Suppose that there are a set of random variables and a set of sigma algebras where both sets are indexed by the same set endowed with a neighborhood system. When the set of random variables satisfies CND, any two non-adjacent sets of random variables are conditionally independent given sigma algebras having indices in one of the two sets’ neighborhood. Random variables with CND include those with conditional dependency graphs and a class of Markov random fields with a global Markov property. The CND property is useful for modeling cross-sectional dependence governed by a complex, large network. This paper provides two main results. The first result is a stable central limit theorem for a sum of random variables with CND. The second result is a Donsker-type result of stable convergence of empirical processes indexed by a class of functions satisfying a certain bracketing entropy condition when the random variables satisfy CND.

Original languageEnglish (US)
Pages (from-to)1189-1224
Number of pages36
JournalBernoulli
Volume25
Issue number2
DOIs
StatePublished - May 2019

Keywords

  • Conditional neighborhood dependence
  • Dependency graphs
  • Empirical processes
  • Markov random fields
  • Maximal inequalities
  • Stable central limit theorem

ASJC Scopus subject areas

  • Statistics and Probability

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