Some Tauberian Theorems Related to Coin Tossing

Abstract

Let $A$ be a subset of the integers and let $S_n$ be the number of heads in $n$ tosses of a $p$ coin. If $\lim_{n\rightarrow\infty} P(S_n \in A)$ exists for some $p$ then the limit exists for all $p$ and does not depend on $p$. The relation of the limit to the density of $A$ and to a similar Poisson limit is also given.