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Increasing subsequences of random walks

Published online by Cambridge University Press:  23 September 2016

OMER ANGEL ,
RICHÁRD BALKA  and
YUVAL PERES
OMER ANGEL
Affiliation:
University of British Columbia, Vancouver BC, V6T 1Z2, Canada. e-mail: angel@math.ubc.ca
RICHÁRD BALKA
Affiliation:
University of British Columbia, and Pacific Institute for the Mathematical Sciences, Vancouver BC, V6T 1Z2, Canada. e-mail: balka@math.ubc.ca
YUVAL PERES
Affiliation:
Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, U.S.A. e-mail: peres@microsoft.com
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Abstract

Given a sequence of n real numbers {Si}i⩽n, we consider the longest weakly increasing subsequence, namely i1 < i2 < . . . < iL with Sik ⩽ Sik+1 and L maximal. When the elements Si are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that ${\mathbb E} L=(2+o(1)) \sqrt{n}$.

We consider the case when {Si}i⩽n is a random walk on ℝ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies ${\mathbb E} L\geq c\sqrt{n}$. Our main result is an upper bound ${\mathbb E} L\leq n^{1/2 + o(1)}$, establishing the leading asymptotic behavior. If {Si}i⩽n is a simple random walk on ℤ, we improve the lower bound by showing that ${\mathbb E} L \geq c\sqrt{n} \log{n}$.

We also show that if {Si} is a simple random walk in ℤ2, then there is a subsequence of {Si}i⩽n of expected length at least cn1/3 that is increasing in each coordinate. The above one-dimensional result yields an upper bound of n1/2+o(1). The problem of determining the correct exponent remains open.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 1 , July 2017 , pp. 173 - 185
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

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