Bias and Division in the Free World

Sampling bias is a foundational concept in statistics; associated bias transforms, such as size bias, have come to play important roles in probability theory of late. The first author and G. Reinert introduced zero bias, a transform whose unique fixed point is the normal distribution; it has become a standard tool in Stein's method and Gaussian approximation. Very recently, connections between zero bias and the class of infinitely divisible distributions have been found. In this paper, we develop a free probabilistic analog of the zero bias transform, proving its existence and regularity. The free zero bias has the semicircle law (free probability's central limit distribution) as its unique fixed point. We offer a construction of the free zero bias that mirrors a classical one incorporating square bias with a mollifier, and in the process develop a surprisingly new class of distributional operations through their Cauchy transforms. We then explore connections between the free zero bias, and size bias, with the class of freely infinitely divisible distributions. We develop a new self-contained treatment of the subject, together with a new characterization of free infinite divisibility using bias transforms. We also develop a parallel treatment of positively freely infinitely divisible distributions, which can also be characterized by a new kind of Levy--Khintchine formula that has no known classical analogue, and we use this to both give several new descriptions of such distributions and furnish new examples using these bias methods.