The Empirical Distribution of a Large Number of Correlated Normal Variables

David Azriel, Armin Schwartzman

Research output: Contribution to journalArticlepeer-review


Motivated by the advent of high-dimensional, highly correlated data, this work studies the limit behavior of the empirical cumulative distribution function (ecdf) of standard normal random variables under arbitrary correlation. First, we provide a necessary and sufficient condition for convergence of the ecdf to the standard normal distribution. Next, under general correlation, we show that the ecdf limit is a random, possible infinite, mixture of normal distribution functions that depends on a number of latent variables and can serve as an asymptotic approximation to the ecdf in high dimensions. We provide conditions under which the dimension of the ecdf limit, defined as the smallest number of effective latent variables, is finite. Estimates of the latent variables are provided and their consistency proved. We demonstrate these methods in a real high-dimensional data example from brain imaging where it is shown that, while the study exhibits apparently strongly significant results, they can be entirely explained by correlation, as captured by the asymptotic approximation developed here. Supplementary materials for this article are available online.

Original languageEnglish
Pages (from-to)1217-1228
Number of pages12
JournalJournal of the American Statistical Association
Issue number511
StatePublished - 3 Jul 2015


  • Asymptotic approximation
  • Dependent random variables
  • Empirical null
  • Factor analysis
  • High-dimensional data
  • Strong correlation

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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