Tauberian theorem

On the other hand, it is straightforward to show, using variational arguments, that E(X) is a monotonic decreasing function of X > Xc. Then the Tauberian theorem for differentiation of asymptotic expression [46] holds, and from Eq. (13) we obtain... 

This Tauberian theorem is as follows. Suppose that... 

For a proof of Theorem A.2 see [Bingham et al. (1987), Theorem 1.7.1 and Corollary 8.1.7]. We stress however that the two theorems to which we have just referred, called Tauberian theorem, give also the converse implications and Tauberian precisely refers to the converse statements. Theorem A.2 just gives the asymptotic behavior of the Laplace transform, given the asymptotic behavior of the function and it is appropriate to refer to it as an Abelian theorem. 

The proof of Theorem A. 11 is based on the fact that the local limit expansion yields the asymptotic behavior of the Laplace transform of P(5 = 0), which in turn, via formula (A.42), yields the asymptotic behavior of the Laplace transform of the distribution of r. A Tauberian argument then gives directly the asymptotic behavior of P(r > n), but extracting the asymptotic behavior of P(t = n) is not obvious. 

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