Exponential Combinatorial Extrema

Abstract

We study the following type of problem. For each K we have a family { X ^K _i : i ∈ I _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