The Littlewood-Offord problem for Markov chains

Shravas Rao

doi:10.1214/21-ecp410

Abstract

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable ε1v1+⋯+εnvn lies in the Euclidean unit ball, where ε1,…,εn∈{−1,1} are independent Rademacher random variables and v1,…,vn∈Rd are fixed vectors of at least unit length. We extend some known results to the case that the εi are obtained from a Markov chain, including the general bounds first shown by Erdős in the scalar case and Kleitman in the vector case, and also under the restriction that the vi are distinct integers due to Sárközy and Szemeredi. In all extensions, the upper bound includes an extra factor depending on the spectral gap and additional dependency on the dimension. We also construct a pseudorandom generator for the Littlewood-Offord problem using similar techniques.

Highlights

Let v1, . . . , vn ∈ Rd be fixed vectors of Euclidean length at least 1, and let ε1, . . . , εn be independent Rademacher random variables, so that Pr[εi = 1] = Pr[εi = −1] = 1/2 for all i
The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable ε1v1 + · · · + εnvn lies in the Euclidean unit ball, where ε1, . . . , εn ∈ {−1, 1} are independent Rademacher random variables and v1, . . . , vn ∈ Rd are fixed vectors of at least unit length
We extend some known results to the case that the εi are obtained from a Markov chain, including the general bounds first shown by Erdos in the scalar case and Kleitman in the vector case, and under the restriction that the vi are distinct integers due to Sárközy and Szemeredi

Summary

Introduction

There exists a Berry-Esseen theorem for Markov chains [13] and various concentration inequalities for Markov chain [4, 10, 9] In all of these cases, there is an extra factor in the bounds in terms of λ which disappears if λ = 0. We obtain the following theorem that upper bounds the probability that the random sum is concentrated on any unit ball. One interpretation of Theorem 1.4 is that one can obtain similar results as in the Littlewood-Off√ord problem in one dimension using much less randomness, and in particular, using C1 n bits of randomness rather than n This setting was considered in [7], in which the authors were able to construct an explicit set of cardinality n2nc , from which a random sample satisfies log(n)C1/c. Sampling from the set in Theorem 1.4 guarantees a stronger bound on the probability that the sum lands in any interval, while requiring more randomness when c < 1/2

Future work

Preliminaries

The Littlewood-Offord problem for random variables from a Markov chain

Extension to distinct vi’s

A pseudorandom generator for the Littlewood-Offord problem