SCHWEITZER 1978 Problem 10

Let $Y_n$ be a binomial random variable with parameters $n$ and $p$ . Assume that a certain set $H$ of positive integers has a density and that this density is equal to $d$ . Prove the following statements:

(a) $\lim _{n \rightarrow \infty}P(Y_n\in H)=d$ if $H$ is an arithmetic progression.

(b) The previous limit relation is not valid for arbitrary $H$ .

(c) If $H$ is such that $P(Y_n \in H)$ is convergent, then the limit must be equal to $d$ .

L. Posa