Stein's method via approximate zero biasing and positive association with applications to combinatorial central limit theorem and statistical physics

2018-10-15Stein's method is nowadays one of the most powerful methods to prove limit theorems in probability and statistics. It was first introduced in 1972 by Charles Stein and used originally for normal approximation. Its exceptional feature over other methods is that it is based on a characteristic equation of the normal distribution and uses coupling constructions. In particular, it does not require the use of complex valued characteristic functions. Since its inception the method has grown dramatically and has now expanded to cover other distributional approximations and even is applied in non-distributional approximation contexts such as concentration of measure. Two additional standout features of the method is that it provides non asymptotic bounds on the distance between distributions for finite sample sizes, and that it can handle a number of situations involving dependence. ❧ In 1997, Larry Goldstein and Gesine Reinert introduced a coupling technique using the zero bias transformation that can be applied to the combinatorial central limit theorem in the case where the given random permutation has the uniform distribution. Our first main focus of this dissertation is to generalize the zero bias coupling to approximate zero bias couplings, and also to extend the application of the combinatorial central limit theorem to the case where the random permutation has the Ewens distribution with parameter θ> 0, which contains the uniform distribution as a special case when θ= 1. ❧ Stein's method also has many well-known applications related to statistical physics. Positive association among the underlying variables is often satisfied for various models in the field. Our second main focus is to develop an L¹ version of Stein's method for sums of positively associated random variables and provide results in the multidimensional case where the L¹ metric is replaced by a smooth functions metric. These results are then applied to four well-known models in statistical physics and particle systems; the ferromagnetic nearest-neighbor Ising model, the supercritical bond percolation model, the voter model and the contact process