Exponential Combinatorial Extrema

  • David Aldous
Part of the Applied Mathematical Sciences book series (AMS, volume 77)

Abstract

We study the following type of problem. For each K we have a family { X i K : iI K } of random variables which are dependent but identically distributed; and |I K | → ∞ exponentially fast as K → ∞. We are interested in the behavior of \( {M_K} = {\kern 1pt} {\max _{i \in {I_K}}}{\kern 1pt} X_i^K \). Suppose that there exists c* ∈ (0,∞) such that (after normalizing the X’s, if necessary)
$$ \begin{array}{*{20}{c}} {\left| {{I_K}} \right|P(X_i^K > c) \to 0}&{as{\kern 1pt} K \to \infty }&{all{\kern 1pt} c > c * } \\ {\left| {{I_K}} \right|P(X_i^K > c) \to \infty }&{as{\kern 1pt} K \to \infty }&{all{\kern 1pt} c > c * } \end{array} $$
Then Boole’s inequality implies
$$ P({M_k} > c) \to 0asK \to \infty ;allc > c* $$
(Gla)
Call c* the natural outer bound for M K (for a minimization problem the analogous argument gives a lower bound c)*; we call these outer bound for consistency

Keywords

Random Graph Root Vertex Offspring Distribution Clump Size Large Deviation Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • David Aldous
    • 1
  1. 1.Department of StatisticsUniversity of California-BerkeleyBerkeleyUSA

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