On minimal-moment generating functions (Q1424674)

For a distribution function \(F\) on \(R_+\) a minimal-moment generation function is defined as \(\varphi(z)=\sum_{k\geq 1}z^{k-1}m_k\), where \(m_k=\mathbf{E}[\min_{1\leq j\leq k} X_j]\), \(X_j\) are i.i.d. with d.f. \(F\). It is shown that under mild conditions \[ F^{-1}\left(1-{1\over \omega+\varepsilon}\right)- F^{-1}\left(1-{1\over \omega-\varepsilon}\right) ={1\over 2\pi i}\oint\varphi(z)dz, \] where the integral is taken over \(\{z:\;| z-\omega| =\varepsilon\}\). It is also demonstrated that the weak convergence of uniformly integrable r.v.s is equivalent to the uniform convergence of their \(\varphi(z)\) on \(| z| <1\).