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Maximally dependent random variables

T L Lai1, H Robbins

  • 1Department of Mathematical Statistics, Columbia University, New York, N.Y. 10027.

Proceedings of the National Academy of Sciences of the United States of America
|February 1, 1976
PubMed

Insights

This study analyzes the maximum of n random variables, establishing bounds for its expected value. The expected maximum is equal to a specific bound under maximal dependence and approximates the independent case for large n.

Area of Science:

  • Probability Theory
  • Statistics
  • Extreme Value Theory

Background:

  • Examines the distribution of the maximum of n random variables (M(n)) with a common marginal distribution function (F).
  • Introduces the concept of 'maximal dependence' to define a specific relationship between the random variables.

Purpose of the Study:

  • To establish theoretical bounds for the expected value of the maximum of n random variables (EM(n)).
  • To investigate the conditions under which EM(n) reaches its upper bound, specifically under maximal dependence.
  • To compare the expected maximum with the independent case as n approaches infinity.

Main Methods:

  • Derivation of an inequality for the expected maximum: EM(n) <= m(n).
  • Definition and application of 'maximal dependence': P(M(n) > x) = min{1, n[1 - F(x)]}.
  • Asymptotic analysis as n approaches infinity, comparing EM(n) with the expected maximum under independence (m(n)*).

Main Results:

  • EM(n) is shown to be less than or equal to a defined bound m(n).
  • Equality EM(n) = m(n) holds under the condition of maximal dependence.
  • For large n, EM(n) asymptotically approaches m(n) and m(n)* (expected maximum for independent variables), given specific conditions on F.

Conclusions:

  • The study provides precise bounds for the expected maximum of random variables.
  • Maximal dependence represents a specific scenario where the upper bound for the expected maximum is achieved.
  • The behavior of the expected maximum converges towards the independent case under certain distributional assumptions and large sample sizes.

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