L ^∞ by Bounded Couplings

Abstract

The theme of Chap. 5 is to provide upper bounds in the L ^∞, or Kolmogorov distance, that can be applied when certain bounded couplings can be constructed between an auxiliary random variable $\widetilde{W}$ and the variable W. Important cases considered include when the variable $\widetilde{W}$ has the same distribution as W, or has the zero bias or size bias distribution of W. The bounds shown in this chapter are often interpretable, sometimes directly, as a distance between W and $\widetilde{W}$, a small bound being a reflection of a small distance. The chapter concludes with the use of smoothing inequalities to obtain distances between W and the normal over general function classes, one special case being the derivation of Kolmogorov distance bounds when bounded size bias couplings exist.