PC Seminar: Bias and Division in the Free World

Abstract: Given $\sigma^2 \in (0,\infty)$, a theorem of Kolmogorov states that $X \in \mathbb{ID}_{0,\sigma^2}$, the set of all infinitely divisible random variables with mean zero and variance $\sigma^2 \in (0,\infty)$, if and only if there exists a probability measure $\nu$ on $\mathbb{R}$ such that the characteristic function $\varphi$ of $X$ satisfies

\begin{align} \label{eq:logphi.zero}

\phi(t) = \exp\left(-\frac{\sigma^2 t^2}{2}\nu(\{0\})

\sigma^2+\int_{\mathbb{R} \setminus\{0\}}

\frac{e^{itx}-1-itx}{x^2}\,\nu(dx)

\right),\quad t\in \mathbb{R}.

\end{align}

From Stein's method, for every mean zero, variance $\sigma^2$ random variable $X$ there exists a unique `$X$-zero bias' distribution $\mathcal{L}(X^*)$ such that

$$

E[Xf(X)]=\sigma^2 E[f'(X^*)] \quad \mbox{for all Lipschitz$_1$ functions $f$.}

$$

The mapping $\mathcal{L}(X) \rightarrow \mathcal{L}(X^*)$ has the Gaussian $\mathcal{N}(0,\sigma^2)$ distribution as its unique fixed point. Using probabilistic techniques, we show that $X \in \mathbb{ID}_{0,\sigma^2}$ if and only if

\begin{align*} %\label{eq:increment}

X^*=_d X+UY

\end{align*}

where $=_d$ denotes equality in distribution, with $X,U,Y$ independent and $U \sim \mathcal{U}[0,1]$,

Similarly, in free probability, we show that for all mean zero, variance $\sigma^2 \in (0,\infty)$ random variables there exists a unique distribution $X^\circ$ such that

$$

E[Xf(X)]=\sigma^2 E[f'(UX^\circ+(1-U)Y^\circ)] \quad \mbox{for all Lipschitz$_1$ functions $f$,}

$$

where $Y^\circ=_d X^\circ$, the variables $X^\circ, Y^\circ, U$ are independent, and $U \sim \mathcal{U}[0,1]$. The mapping $\mathcal{L}(X) \rightarrow \mathcal{L}(X^\circ)$ has the $\mathcal{S}(0,\sigma^2)$ semi-circle distribution as its unique fixed point, and $X \in \mathbb{FID}_{0,\sigma^2}$, the set of all freely infinitely divisible random variables with mean zero and variance $\sigma^2 \in (0,\infty)$, if and only if there exists a random variable $Y$ such that, with $G_W$ denoting the Cauchy transform of $W$,

$$

G_{X^\circ}(z)=G_{Y^\sharp}(1/G_X(z)) \quad \mbox{where} \quad G_{Y^\sharp}(z)=\sqrt{G_Y(z)/z}.

$$

These new identities lead to probabilistic interpretations of the corresponding L\'evy measures, such as $\nu$ in \eqref{eq:logphi.zero} in the classic case.

This is a seminar in our seminar series on Probability and Combinatorics (PC).