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Some Tauberian Theorems Related to Coin Tossing

Persi Diaconis and Charles Stein
The Annals of Probability
Vol. 6, No. 3 (Jun., 1978), pp. 483-490
Stable URL: http://www.jstor.org/stable/2243150
Page Count: 8
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Some Tauberian Theorems Related to Coin Tossing
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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.

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