Exponential Combinatorial Extrema

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


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
$$ P({M_k} > c) \to 0asK \to \infty ;allc > c* $$
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


Random Graph Root Vertex Offspring Distribution Clump Size Large Deviation Theorem 
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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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