We present, in a unified way, a Stein methodology for infinitely divisible laws (without Gaussian component) having finite first moment. Based on a correlation representation, we obtain a characterizing non-local Stein operator which boils down to classical Stein operators in specific examples. Thanks to this characterizing operator, we introduce various extensions of size bias and zero bias distributions and prove that these notions are closely linked to infinite divisibility. Combined with standard Fourier techniques, these extensions also allow obtaining explicit rates of convergence for compound Poisson approximation in particular towards the symmetric α-stable distribution. Finally, in the setting of non-degenerate self-decomposable laws, by semigroup techniques, we solve the Stein equation induced by the characterizing non-local Stein operator and obtain quantitative bounds in weak limit theorems for sums of independent random variables going back to the work of Khintchine and Lévy.
We provide a bound on a natural distance between finitely and infinitely supported elements of the unit sphere of ℓ 2 (IN ⋆ ), the space of real valued sequences with finite ℓ 2 norm. We use this bound to estimate the 2-Wasserstein distance between random variables which can be represented as linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure which is related to Nourdin and Peccati's Malliavin-Stein method. The main area of application of our results is towards the computation of quantitative rates of convergence towards elements of the second Wiener chaos. After particularizing our bounds to this setting and comparing them with the available literature on the subject (particularly the Malliavin-Stein method for variance-gamma random variables), we illustrate their versatility by tackling three examples: chi-squared approximation for second order U -statistics, asymptotics for sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.
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