2 research outputs found
A Note on the Probability of Rectangles for Correlated Binary Strings
Consider two sequences of independent and identically distributed fair
coin tosses, and , which are
-correlated for each , i.e. .
We study the question of how large (small) the probability can be among all sets of a given cardinality.
For sets 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 . 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 in
the regime of . We also prove a similar tight lower bound, i.e.
show that for the pair of opposite Hamming balls approximately
minimizes the probability