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Distributional transformations and coupling techniques used in Stein’s method have also found applications in waiting-time paradoxes, tightness, Palm mea- sures, analysis of the lognormal distribution, Skorohod embedding, concen- tration of measure, infinite divisibility, number theory, statistical robustness and sampling methods.

Here we explore its connection to classical and free infinite divisibility using the ‘X-zero bias’ distribution L(X∗), which exists uniquely for every mean zero, variance σ2 random variable, and has characterizing equation

E[Xf(X)] = σ2E[f′(X∗)] for all Lipschitz1 functions f.

The mapping L(X) → L(X∗) has the Gaussian N(0,σ2) distribution as its unique fixed point. Using probabilistic techniques, we show that every mean zero X with finite, non-zero variance is infinitely divisible if and only if

X∗ =d X+UY

where =d denotes equality in distribution, with X, U, Y independent and U ∼ U[0,1],

Similarly, in free probability, we show that for all mean zero, variance σ2 ∈ (0, ∞) random variables there exists a unique distribution X◦ such that

E[Xf(X)] = σ2E[f′(UX◦ + (1 − U)Y ◦)] for all Lipschitz1 functions f,

where Y ◦ =d X◦, the variables X◦, Y ◦, U are independent, and U ∼ U[0, 1]. The mapping L(X) → L(X◦) has the semi-circle S(0, σ2) distribution as its unique fixed point, and X ∈ FID0,σ2 , the set of all freely infinitely divisible random variables with mean zero and variance σ2 ∈ (0,∞), if and only if there exists a random variable Y such that, with GW denoting the Cauchy transform of W,

GX◦ (z) = GY ♯ (1/GX (z)) where GY ♯ (z) = pGY (z)/z. These new identities lead to probabilistic interpretations of the corresponding

L ́evy measures associated with infinitely divisible random variables.