L ¹ Bounds

Abstract

Chapter 4 develops a theory for obtaining L ¹ bounds using the zero bias coupling which can be applied in non-independent settings. A number of examples are presented for illustration. First considering independent random variables, an L ¹ Berry–Esseen bound is shown, followed by a demonstration of a type of contraction principle ‘toward the normal.’ Bounds in L ¹ are then proved for hierarchical structures, that is, self similar, fractal type objects whose scale at small levels is replicated on the larger. Then, making the first departure from independence, L ¹ bounds are shown for the projections of random vectors having distribution concentrated on regular convex sets in Euclidean space. Next, illustrating a different coupling, L ¹ bounds to the normal for the combinatorial central limit theorem are given. Chatterjee’s L ¹ theorem for functions of independent random variables is presented, and applied to the approximation of the distribution of the volume covered by randomly placed spheres in the Euclidean torus. Results are then given for sums of locally dependent random variables, with applications including the number of local maxima on a graph. The chapter concludes with a consideration of a class of smooth functions for which convergence to the normal is at an accelerated rate subject to additional moment assumptions.