Optimally revealing bits for rejection sampling

Rejection sampling is a popular method used to generate numbers that follow some given distribution. We study the use of this method to generate random numbers in the unit interval from increasing probability density functions. We focus on the problem of sampling from $n$ correlated random variables from a joint distribution whose marginal distributions are all increasing. We show that, in the worst case, the expected number of random bits required to accept or reject a sample grows at least linearly and at most quadratically with $n$.