A limit obtained from a probability distribution on the positive integers

This is proved in Spitzer's "Principles of Random Walk", claim P3 of Section 9 (p.100 of the second edition).

This result belongs to P. Erdös, W. Feller, and H. Pollard (1949). It worth mention that if $\sum a_n p_n<\infty$, this follows from Wiener division theorem in the algebra of absolutely summable Fourier series (which has a famous Banach algebra proof proposed by Gelfand as the application of maximal ideals in Banach algebras theory): $$f(x):=\frac{1-\sum p_nx^n}{1-x}=\sum_n p_n(1+x+\dots+x^{n-1})$$ is absolutely summable and satisfies $f(e^{i\alpha})\ne 0$ for all real $\alpha$ (that is equivalent that no $d>1$ divides all $n$ with $p_n\ne 0$), by Wiener theorem it implies that $1/f=\sum (a_k-a_{k-1})x^k$ is also absolutely summable power series and $\sum_k (a_k-a_{k-1})=\lim a_k=1/f(1)=1/\sum np_n$.

According to Korevaar's book on Tauberian theory (2004), the analytic proof of the $\sum np_n=\infty$ case is still open. Korevaar refers to W. Feller (An introduction to probability theory and its applications, vol. 1, Chapter 15) and G. Grimmet and D. Stirzaker (Probability and random processes, 1992) for the probabilistic proof covering the cases both of finite and infinite expectation.