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VOL. 5 | 2009 On the longest increasing subsequence for finite and countable alphabets
Christian Houdré, Trevis J. Litherland

Editor(s) Christian Houdré, Vladimir Koltchinskii, David M. Mason, Magda Peligrad

Abstract

Let X1, X2, …, Xn, … be a sequence of iid random variables with values in a finite ordered alphabet {α1, …, αm}. Let LIn be the length of the longest increasing subsequence of X1, X2, …, Xn. Properly centered and normalized, the limiting distribution of LIn is expressed as various functionals of m and (m−1)-dimensional Brownian motions. These expressions are then related to similar functionals appearing in queueing theory, allowing us to further describe asymptotic behaviors when, in turn, m grows without bound. The finite alphabet results are then used to treat the countable (infinite) alphabet case.

Information

Published: 1 January 2009
First available in Project Euclid: 2 February 2010

zbMATH: 1243.60022

Digital Object Identifier: 10.1214/09-IMSCOLL513

Subjects:
Primary: 05A16 , 60C05 , 60F05 , 60F17 , 60G15 , 60G17

Keywords: Brownian functional , functional central limit theorem , Longest increasing subsequence , Tracy-Widom distribution

Rights: Copyright © 2009, Institute of Mathematical Statistics

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