TV over Bernoulli products: the small parameter regime

Abstract

We study the total variation distance (TV) between two $n$-fold Bernoulli product measures parametrized by $\vec{p}=(p_1,\dots ,p_n)$ and $\vec{q}=(q_1,\dots ,q_n)$, respectively, in the \emph{tiny} and \emph{small} regimes. In the tiny regime, we have $p_i,q_i\lesssim 1/n^2$, and in the small regime, $p_i,q_i\lesssim 1/n$. We discover that in the tiny regime, the TV distance behaves as $\Vert \vec{p}-\vec{q}{\Vert }_1$, while in the small regime, it behaves as \[ \sum_{i=1}^n \Big| p_i\prod_{j\neq i}(1-p_j) - q_i\prod_{j\neq i}(1-q_j) \Big|, \] both up to absolute constants. Along the way we discover some identities of possible independent interest.