Let $f(x)$ be a bilateral Laplace transform of a measure $\mu$: $$ f(x)=\int_{-\infty}^{+\infty} e^{-xt} d\mu(y). $$ Suppose that $f(x)$ converges absolutely in $(a,b)$, and $(a,b)$ do not contain the origin. It is always true that $f(x)$ is analytic in $(a,b)$? Or it is true just for finite measure $\mu$?
Moreover, if the measure $\mu$ is finite, then $(a,b)$ must contain the origin?
Thank you!
The following references investigate the Moment Generating Function (finite measure):
There is a classical reference for the case where the interval $(a,b)$ contains the origin and a recent extension to the case of a general open interval $(a,b)$. This paper that gives a nice overview and proves the converse to Curtiss's theorem in the general case.
Refs:
Curtiss, J.H., 1942. A note on the theory of moment generating functions, 1, Ann. of Math. Statist, 13 (4) : 430-433.
Mukherjea, A., Rao, M. and Suen, S., 2006. A note on moment generating functions, Statistics & Probability Letters, 76 : 1185-1189.
P. Chareka. \The converse to Curtiss theorem for one-sided moment generating functions" (2008). arXiv: 0807.3392 [quant-ph].