2019/10/17 by Kwaśnicki, Mateusz, Simon, Thomas
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1910.07752
A non-negative function f is said to be 'bell-shaped' if f tends to zero at ± ∞ and the n-th derivative of f changes its sign n times for every n = 0, 1, 2, … We provide a complete characterisation of the class of bell-shaped functions: we prove that every bell-shaped function is a convolution of a 'Pólya frequency function' and an *absolutely monotone-then-completely monotone* function. An equivalent condition in terms of the holomorphic extension of the Fourier transform is also given. As a corollary, various properties of bell-shaped functions follow. In particular, we prove that bell-shaped probability distributions are infinitely divisible, and that the zeroes of the n-th derivative of a bell-shaped function grow at a linear rate as n → ∞.