On the longest increasing subsequence for finite and countable alphabets

Abstract

Let X₁, X₂, …, X_n, … be a sequence of iid random variables with values in a finite ordered alphabet {α₁, …, α_m}. Let LI_n be the length of the longest increasing subsequence of X₁, X₂, …, X_n. Properly centered and normalized, the limiting distribution of LI_n 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.