Exponential Combinatorial Extrema
Chapter
Abstract
We study the following type of problem. For each K we have a family { X i K : 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) Then Boole’s inequality implies 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
$$
\begin{array}{*{20}{c}}
{\left| {{I_K}} \right|P(X_i^K > c) \to 0}&{as{\kern 1pt} K \to \infty }&{all{\kern 1pt} c > c * } \\
{\left| {{I_K}} \right|P(X_i^K > c) \to \infty }&{as{\kern 1pt} K \to \infty }&{all{\kern 1pt} c > c * }
\end{array}
$$
$$
P({M_k} > c) \to 0asK \to \infty ;allc > c*
$$
(Gla)
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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© Springer Science+Business Media New York 1989