On the Longest Increasing Subsequence for Finite and Countable Alphabets

Details On the Longest Increasing Subsequence for Finite and Countable Alphabets On the Longest Increasing Subsequence for Finite and Countable Alphabets

2006-12-13

arxiv.org/abs/math/0612364v1

Mathematics

Probability

Scientific paper

Let $X_1, X_2, ..., X_n, ... $ be a sequence of iid random variables with values in a finite alphabet $\{1,...,m\}$. Let $LI_n$ be the length of the longest increasing subsequence of $X_1, X_2, ..., X_n.$ We express the limiting distribution of $LI_n$ as 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 establish asymptotic behaviors as $m$ grows. The finite alphabet results are then used to treat the countable (infinite) alphabet.

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