A Note on the Probability of Rectangles for Correlated Binary Strings
Name
1909.01221.pdf
Description
Submitted version
Size
336.08 KB
Format
Adobe PDF
Checksum (MD5)
1a0d1b2295c2f0c4a26ab6d8bf5c4757
Author(s)
Ordentlich, Or
Polyanskiy, Yury
Shayevitz, Ofer
Date Issued
2020
Journal
IEEE Transactions on Information Theory
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Version
Original manuscript
Abstract
© 1963-2012 IEEE. Consider two sequences of ${n}$ independent and identically distributed fair coin tosses, ${X}=({X}_{1},\ldots,{X}_{n})$ and ${Y}=({Y}_{1},\ldots,{Y}_{n})$ , which are $\rho $ -correlated for each ${j}$ , i.e. $\mathbb {P}[{X}_{j}={Y}_{j}] = {\frac{1+\rho }{ 2}}$. We study the question of how large (small) the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ can be among all sets ${A},{B}\subset \{0,1\}^{n}$ of a given cardinality. For sets $|{A}|,|{B}| = \Theta (2^{n})$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of $|{A}|,|{B}| = 2^{\Theta ({n})}$. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$ in the regime of $\rho \to 1$. We also prove a similar tight lower bound, i.e. show that for $\rho \to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb {P}[{X} \in {A}, {Y}\in {B}]$.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1109/TIT.2020.3018232
