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Bias and Division in the Free World

Larry Goldstein, Todd Kemp

Abstract

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.

Bias and Division in the Free World

Abstract

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.
Paper Structure (35 sections, 45 theorems, 280 equations, 7 figures)

This paper contains 35 sections, 45 theorems, 280 equations, 7 figures.

Table of Contents

  1. Introduction
  2. (Free) Zero Bias and (Free) Stein Kernels
  3. A Free Probability Primer
  4. Noncommutative Probability Spaces
  5. Free Independence
  6. Cauchy Transform
  7. The Voiculescu Transform and the $R$-transform
  8. Free Poisson Distributions
  9. Subordination and Free Conditional Expectation
  10. Free Infinite Divisibility
  11. Positive Free Infinite Divisibility
  12. Existence and properties of the Free Zero Bias and other Transformations
  13. Geometric Mean Transformation
  14. The Size Bias $s$, Square Bias $\Box$ and El Gordo $\flat$ Transformations
  15. Construction of the Free Zero Bias $\circ$
  16. ...and 20 more sections

Key Result

Theorem 1.1

Given any $\sigma^2>0$ and probability measure $\mu$ in $\mathcal{D}_{0,\sigma^2}$, there is a unique probability measure $\mu^\circ$ on $\mathbb{R}$ that satisfies Equation char.freezb with $X\stackrel{\hbox{$\mathrm d$}}= \mu$ and $X^\circ\stackrel{\hbox{$\mathrm d$}}= \mu^\circ$. This measure is

Figures (7)

  • Figure 2: Evaluation of $\partial g(x,y)$ on the partition of the plane induced by the horizontal and vertical lines $y=a,y=b$ and $x=a,x=b$ for $a<b$.
  • Figure 3: The density of the free zero bias of the three point distribution in \ref{['e.3point.meas']}.
  • Figure 4: The density \ref{['e.density.Levy.semicircle']} of the freely infinitely divisible law with mean $0$, variance $1$, and free Lévy--Khintchine measure $\mathcal{L}(V_0)$ a semicircle of variance $t=0.1$ on the left and $t=1$ on the right.
  • Figure 5: The density in \ref{['e.cubic.density']} of the $\boxplus$-infinitely divisible 'Azadi Tower' distribution with Rademacher $Free-$ Lévy measure.
  • Figure 6: The density in \ref{['e.Cauchy.density']} of the mean $0$, variance $\sigma^2=1$, freely infinitely divisible distribution with Cauchy $Free-$ Lévy measure.
  • ...and 2 more figures

Theorems & Definitions (109)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 1.4
  • Remark 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 99 more