关于相关二元串矩形概率的一个注记
A Note on the Probability of Rectangles for Correlated Binary Strings
摘要：
考虑两个$n$独立且同分布的公平硬币投掷序列$X=(X_1,\ldots,X_n)$和$Y=(Y_1,\ldots,Y_n)$，它们对于每个$j$是$ρ$相关的，即$\mathbb{P}[X_j=Y_j] = {1+ρ\over 2}$。我们研究的问题有多大（小）的概率$\mathbb{P}[X \in A, Y\in B]$可以在一个给定的基数的所有集合$A,B\subset\{0,1\}^n$。对于集合$|A|,|B| = Θ(2^n)$，众所周知，最大（最小）概率近似由同心（反同心）汉明球获得，这可以通过超压缩不等式（反向超压缩性）来证明。这里我们考虑$|A|,|B| = 2^{Θ(n)}$的情况。通过应用Polyanskiy-Samorodnitsky（J. Functional Analysis，2019）的超压缩不等式的最新扩展，我们证明了相同大小的Hamming球在$ρ\to 1$的状态下近似最大化$\mathbb{P}[X \in A, Y\in B]$。我们还证明了一个类似的紧下界，即表明，$ρ\to 0$的对相反的汉明球近似最小化概率$\mathbb{P}[X \in A, Y\in B]$。
英文摘要：
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 $ρ$-correlated for each $j$, i.e. $\mathbb{P}[X_j=Y_j] = {1+ρ\over 2}$. 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 $|A|,|B| = Θ(2^n)$ it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive 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 of the same size approximately maximize $\mathbb{P}[X \in A, Y\in B]$ in the regime of $ρ\to 1$. We also prove a similar tight lower bound, i.e. show that for $ρ\to 0$ the pair of opposite Hamming balls approximately minimizes the probability $\mathbb{P}[X \in A, Y\in B]$.
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