Central moment inequalities using Stein's method

TL;DR

This paper develops explicit bounds for central moments of random variables with Stein couplings, facilitating easier analysis of their distributional properties and applications to concentration inequalities and random graph statistics.

Contribution

It introduces a novel approach to bounding central moments using Stein's method, applicable to various dependence structures and random graph models.

Findings

Derived explicit bounds for central moments using Stein couplings.

Bounded moments are closely approximated by those of a normal distribution.

Applied bounds to sums of neighborhood statistics in sparse Erdős–Rényi graphs.

Abstract

We derive explicit central moment inequalities for random variables that admit a Stein coupling, such as exchangeable pairs, size--bias couplings or local dependence, among others. The bounds are in terms of moments (not necessarily central) of variables in the Stein coupling, which are typically local in some sense, and therefore easier to bound. In cases where the Stein couplings have the kind of behaviour leading to good normal approximation, the central moments are closely bounded by those of a normal. We show how the bounds can be used to produce concentration inequalities, and compare them to those existing in related settings. Finally, we illustrate the power of the theory by bounding the central moments of sums of neighbourhood statistics in sparse Erd\H{o}s--R\'enyi random graphs.

Equations

$$\text{mathbbm}E[G(f(W^{\prime })-f(W))]=\text{mathbbm}E[(W-\mu )f(W)]+\text{mathbbm}E[Rf(W)],$$

$$|\text{mathbbm}E[(W-\mu )f(W)-{\sigma }^2{f}^{\prime }(W)]|\le \epsilon ({\sigma }^2\Vert f^{\prime }{\Vert }_{\infty }+{\sigma }^3\Vert f^{\prime \prime }{\Vert }_{\infty })$$

$$d_{\mathrm{BW}}(P,Q)\ :=\ \sup_{h\in{\mathcal{H}}_{\mathrm{BW}}}\Bigl{\lvert}\int h\,dP-\int h\,dQ\Bigr{\rvert},$$

$$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\mathbbm{E}[(W-\mu)f(W)-\sigma^{2}f^{\prime}(W)]\ =\ \mathbbm{E}[G(f(W^{\prime})-f(W))]-\sigma^{2}\mathbbm{E}f^{\prime}(W)$\mbox{}\hfil\\ &\qquad\qquad\qquad\qquad\ =\ \mathbbm{E}[(GD-\sigma^{2})f^{\prime}(W)]+\mathbbm{E}[G(f(W+D)-f(W)-Df^{\prime}(W))].\end{split}$$

$$\bigl{\lvert}\mathbbm{E}[R|W]\bigr{\rvert}\ \leq\ \varepsilon\lvert W-\mu\rvert+T.$$

$$\begin{split}\|W-\mu\|_{2k}&\ \leq\ \frac{A}{1-\varepsilon}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(1+\sqrt{\frac{B(1-\varepsilon)}{A(2k-1)}}}\right)^{2k-1}-1}\right]+\frac{\|T\|_{2k}}{1-\varepsilon}\\ &\ \leq\ \sqrt{\frac{(2k-1)AB}{1-\varepsilon}}\exp\mathopen{}\mathclose{{}\left\{\sqrt{\frac{B(1-\varepsilon)(2k-1)}{A}}}\right\}+\frac{\|T\|_{2k}}{1-\varepsilon}\,.\end{split}$$

$$\begin{array}{rl}\Vert W-\mu {\Vert }_r& \le \sqrt{\frac{2(r-1)\Vert G{\Vert }_r\Vert D{\Vert }_r}{1-\epsilon -{\epsilon }^{\prime }}}\end{array}$$