Increasing subsequences of random walks

TL;DR

This paper investigates the length of the longest increasing subsequence in random walks, establishing asymptotic bounds and exploring higher-dimensional cases, thus extending classical results from i.i.d. sequences.

Contribution

It provides the first asymptotic upper bound for the expected length of the LIS in random walks and improves lower bounds for simple random walks, also exploring multi-dimensional cases.

Findings

Expected LIS length in general random walks is at most n^{1/2 + o(1)}.

Lower bound for simple random walks is at least c√n log n.

In 2D, a subsequence of expected length at least c n^{1/3} exists.

Abstract

Given a sequence of $n$ real numbers $\{S_i\}_{i\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\dots <i_L$ with $S_{i_k} \leq S_{i_{k+1}}$ and $L$ maximal. When the elements $S_i$ 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 $\{S_i\}_{i\leq n}$ is a random walk on $\mathbb{R}$ 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 $\{S_i\}_{i\leq n}$ is a simple random walk on $\mathbb{Z}$, we improve the lower bound by showing that $\mathbb{E} L \geq c\sqrt{n} \log{n}$. We also show that if $\{\mathbf{S}_i\}$ is a simple random walk in $\mathbb{Z}^2$, then there is a subsequence of $\{\mathbf{S}_i\}_{i\leq n}$ of expected length at least $cn^{1/3}$ that is increasing in each coordinate. The above one-dimensional result yields an upper bound of $n^{1/2 + o(1)}$. The problem of determining the correct exponent remains open.