Limits of Ratios of Tails of Measures

Abstract

Suppose $\mu$ is a positive measure on the half-line $\lbrack 0, \infty)$, of total mass $m, \Phi$ is the sum of a power series with nonnegative coefficients which converges at the point $m$, and $\lambda$ is the measure on $\lbrack 0, \infty)$ whose Fourier transform $\hat{\lambda}$ is $\Phi(\hat{\mu})$. The lower limit of the ratios $\lambda(\lbrack s, \infty))/\mu(\lbrack s, \infty))$, as $s\rightarrow\infty$, is compared to the number $\Phi'(m)$, under a variety of conditions.