Tight Probability Bounds with Pairwise Independence

While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several results towards finding tight probability bounds in this direction. Firstly, when $k = 1$, the tightest upper bound on the probability of the union of $n$ pairwise independent events is provided in closed-form for any input marginal probability vector $\mathbf{p} \in [0,1]^n$. This result is then extended to derive the tightest lower bound on the probability of the intersection of $n$ pairwise independent events ($k=n$) for any $\mathbf{p} \in [0,1]^n$. To prove the results, we show the existence of two types of positively correlated Bernoulli random vectors with transformed bivariate probabilities, which is of independent interest. Building on this, we show that the ratio of the Boole union bound and the tight pairwise independent bound is upper bounded by $4/3$ and that the ratio is attained. Applications of the result in correlation gap analysis and distributionally robust bottleneck optimization are discussed. Secondly, for any $k \geq 2$ and input marginal probability vector $\mathbf{p} \in [0,1]^n$, new upper bounds are derived by exploiting ordering of probabilities. Numerical examples are provided to illustrate when the bounds provide improvement over existing bounds. Lastly, we identify instances when the existing and new bounds are tight, for example with identical marginal probabilities.