Characterisation of the class of bell-shaped functions

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A non-negative function f is said to be bell-shaped if f tends to zero at \(\pm \infty \) and the n-th derivative of f changes its sign n times for every \(n = 0, 1, 2, \ldots \) We provide a complete characterisation of the class of bell-shaped functions, resembling Bernstein’s identification of completely monotone functions with Laplace transforms of non-negative measures. More precisely, 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, we find various properties of bell-shaped functions. In particular, we prove that bell-shaped probability distributions are infinitely divisible, and that a random walk \(X_n\) (or a Lévy process \(X_t\)) have bell-shaped distributions if and only if the distribution of \(X_1\) is an extended generalised gamma convolution. We also link the rate of growth of the zeroes of n-th derivative of a bell-shaped function f with numbers p such that \(f + p f'\) is bell-shaped. Our result provides a practical method for checking whether a given distribution is bell-shaped, and it is new even for one-sided distributions.