Some Contributions to Stein's Method and the Theory of Probability Distributions

Abstract

In 1972, Charles Stein introduced a novel approach to deriving distributional approximations in probability theory, allowing one to obtain explicit bounds with respect to a probability metric. Originally developed for normal approximations, Stein's method is applicable to a wide range of distributions and it effectively handles dependence. In this thesis, we contribute to two different areas of Stein's method. These contributions include an extension of Stein's method to a new distribution and the derivation of explicit error bounds for distributional approximations in the Kolmogorov distance. We derive a Stein operator for the product of two correlated normal random variables with non-zero means. This Stein operator contains as special cases several Stein operators existing in the literature, and we apply our Stein operator to obtain distributional properties of the product of two correlated normal random variables. Deriving a Kolmogorov distance bound via Stein's method is often technically difficult. This motivates us to contribute to Stein's method by establishing upper bounds on the Kolmogorov distance between two random variables from a wide range of distributions in terms of the Wasserstein or smooth Wasserstein metrics. We utilize our bounds to derive Kolmogorov distance bounds for multivariate normal, beta and variance-gamma approximations. A special case of the product of two correlated normal random variables with zero means is the variance-gamma distribution, which is widely used in financial modelling and has found recent application as a limiting distribution in probability theory. In this thesis, we contribute to the distributional theory of the variance-gamma distribution by exploring the fundamental properties of the ratio and product of two independent variance-gamma distributed random variables. These properties include the probability density function, cumulative distribution function, quantile function and asymptotic behaviours.

Date of Award	31 Dec 2024
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Robert Gaunt (Supervisor)