For a probability distribution function on the real line, a necessary condition on its infinite divisibility is given which deals with its one-sided asymptotic behavior; the proof is based on properties of characteristic functions which are analytic in the upper (lower) half plane.
"A Necessary Condition on the Infinite Divisibility of Probability Distributions." Ann. Math. Statist. 42 (5) 1681 - 1685, October, 1971. https://doi.org/10.1214/aoms/1177693166