Binomial approximation and large deviations for defaults

Let $D_n$ be the total number of defaults among $n$ banks, and let $p$ be the corresponding asymptotic default probability. Default-count conjecture. The random variable $D_n$ should be approximately binomially distributed with parameters $n$ and $p$, and large-deviation estimates for a binomial distribution should apply to $D_n$. This would permit rare-event probabilities for systemic defaults to be estimated through binomial large-deviation theory. The source does not specify a precise mode of approximation or establish the claimed large-deviation principle.