Seminar at SMU Delhi

January 21, 2015 (Wednesday) , 3:30 PM at Webinar
Speaker: Christian Berg, University of Copenhagen, Denmark
Title: On the determinacy/indeterminacy of the Stieltjes moment sequence $(n!)^c,c>0$
Abstract of Talk
Let $f$ be a non-zero Bernstein function, i.e., $$ f(s)=a+bs+\int_0^\infty \left(1-e^{-xs}\right)\,d\nu(x),\;a,b\ge 0,\nu\ge 0. $$ There exists a uniquely determined product convolution semigroup $(\rho_c)_c>0$ on $(0,\infty)$ such that \begin{equation}\label{eq:1} \int_0^\infty x^n\,d\rho_c(x)=(f(1)\cdots\ldots\cdot f(n))^c,\quad c>0,n=0,1,\ldots, \end{equation} see \cite{B1}. The Stieltjes moment sequence in \eqref{eq:1} is always determinate when $c\le 2$ as an easy consequence of the Carleman criterion. However, for $c>2$, it can be determinate or indeterminate depending on $f$. In fact, in the case $f(s)=s$, where the moment sequence is $(n!)^c$, it was proved in \cite{B1} that the moment sequence is indeterminate. In this case $\rho_c=e_c(t)dm(t)$, where $m$ is Lebesgue measure on the half-line and \begin{equation}\label{eq:2} e_c(t)=\frac{1}{2\pi}\int_{-\infty}^\infty t^{ix-1}\Gamma(1-ix)^c\,dx,\quad t>0. \end{equation} The proof of the indetermincy was quite delicate based on asymptotic formulas for stable distributions due to Skorokhod. In recent work with Jos\'e L\'opez, Spain, \cite{B:L} we have found the asymptotic behaviour of $e_c$ at infinity, viz. \begin{equation}\label{eq:3} e_c(t)\sim \frac{(2\pi)^{(c-1)/2}}{\sqrt{c}}\frac{e^{-ct^{1/c}}}{t^{(c-1)/(2c)}}. \end{equation} From \eqref{eq:3} it is easy to derive the indeterminacy of $e_c$ from a criterion of Krein. In the lecture I will give the necessary background for the above. \begin{thebibliography}{abc} \bibitem{B1} Berg, C., On powers of Stieltjes moment sequences, I. {\it J. Theor. Prob.} {\bf 18} (2005), 871--889. \bibitem{B:L} Berg, C., L{\'o}pez, J. L., Asymptotic behaviour of the Urbanik semigroup. To appear in J. Approx. Theory 2015. \end{thebibliography}