A Note on the Probability of Rectangles for Correlated Binary Strings
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Consider two sequences of n independent and identically distributed fair coin tosses, X=(X_1,...,X_n) and Y=(Y_1,...,Y_n), which are ρ-correlated for each j, i.e. P[X_j=Y_j] = 1+ρ 2. We study the question of how large (small) the probability P[X ∈ A, Y∈ B] can be among all sets A,B⊂{0,1}^n of a given cardinality. For sets |A|,|B| = Θ(2^n) it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be easily proved via the Nelson-Bonami-Beckner hypercontrictive inequality (reverse hypercontractivity). Here we consider the case of |A|,|B| = 2^-Θ(n). By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls approximately maximize P[X ∈ A, Y∈ B] in the regime of ρ→ 1. We also prove a similar tight lower bound, i.e. show that for ρ→ 0 the pair of opposite Hamming balls approximately minimizes the probability P[X ∈ A, Y∈ B].
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