Contributions to Stein's method and some limit theorems in probability
- Author(s): Dey, Partha Sarathi
- Advisor(s): Chatterjee, Sourav
- Evans, Steven N
- et al.
In this dissertation we investigate three different problems related to (1) concentration inequalities using Stein's method of exchangeable pair, (2) first-passage percolation along thin lattice cylinders and (3) limiting spectral distribution of random linear combinations of projection matrices.
Stein's method is a semi-classical tool for establishing distributional convergence, particularly effective in problems involving dependent random variables. A version of Stein's method for concentration inequalities was introduced in the Ph.D.~thesis of Sourav Chatterjee to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures.
In the first part of the dissertation we provide some extensions of the theory and three new applications: (1) We obtain a concentration inequality for the magnetization in the Curie-Weiss model at critical temperature (where it obeys a non-standard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erd\H os-R