Strictly subgaussian probability distributions

We explore probability distributions on the real line whose Laplace transform admits an upper bound of subgaussian type known as strict subgaussianity. One class in this family corresponds to entire characteristic functions having only real zeros in the complex plane. Using Hadamard’s factorization theorem, we extend this class and propose new sufficient conditions for strict subgaussianity in terms of location of zeros of the associated characteristic functions. The second part of this note deals with Laplace transforms of strictly subgaussian distributions with periodic components. This class contains interesting examples, for which the central limit theorem with respect to the Rényi entropy divergence of infinite order holds.