Characteristic Functions

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Subsampled Rényi Differential Privacy and Analytical Moments Accountant (2018, methodology): “For example, if we use a polynomial upper bound of (cid:15) ( · ) , we know from the theory of CGF that no distribution have a CGF of polynomial order higher than 2 and the only distribution that has polynomial order exactly two is the Gaussian distribution (Lukacs, 1970).”
Affine processes are regular (2009, methodology): “Our proof uses the techniques originally presented in Montgomery and Zippin (1955) for continuous transformation groups, and follows in part the proof of (Dawson and Li, 2006, Theorem 4.”
Learning Structural Node Embeddings via Diffusion Wavelets (2017, methodology): “The function φ X ( t ) fully characterizes the distribution of X because it captures information about all the moments of probability distribution X (Lukacs, 1970).”
Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type (2009, background): “This process has Lévy density u(x) = aκ2 κ x−(κ+1) (1 − κ) exp ( −b 1 κ x 2 ) and then it follows by Lukacs (1970), Theorem 5.11.2, that the tempered stable distribution is self-decomposable.”