Title:

Rates of convergence of variancegamma approximations via Stein's method

Stein's method is a powerful technique that can be used to obtain bounds for approximation errors in a weak convergence setting. The method has been used to obtain approximation results for a number of distributions, such as the normal, Poisson and Gamma distributions. A major strength of the method is that it is often relatively straightforward to apply it to problems involving dependent random variables. In this thesis, we consider the adaptation of Stein's method to the class of VarianceGamma distributions. We obtain a Stein equation for the VarianceGamma distributions. Uniform bounds for the solution of the Symmetric VarianceGamma Stein equation and its first four derivatives are given in terms of the supremum norms of derivatives of the test function. New formulas and inequalities for modified Bessel functions are obtained, which allow us to obtain these bounds. We then use local approach couplings to obtain bounds on the error in approximating two asymptotically VarianceGamma distributed statistics by their limiting distribution. In both cases, we obtain a convergence rate of order n^{1} for suitably smooth test functions. The product of two normal random variables has a VarianceGamma distribution and this leads us to consider the development of Stein's method to the product of r independent meanzero normal random variables. An elegant Stein equation is obtained, which motivates a generalisation of the zero bias transformation. This new transformation has a number of interesting properties, which we exploit to prove some limit theorems for statistics that are asymptotically distributed as the product of two central normal distributions. The VarianceGamma and Product Normal distributions arise as functions of the multivariate normal distribution. We end this thesis by demonstrating how the multivariate normal Stein equation can be used to prove limit theorems for statistics that are asymptotically distributed as a function of the multivariate normal distribution. We establish some sufficient conditions for convergence rates to be of order n^{1} for smooth test functions, and thus faster than the O(n^{1/2}) rate that would arise from the BerryEsseen Theorem. We apply the multivariate normal Stein equation approach to prove VarianceGamma and Product Normal limit theorems, and we also consider an application to Friedman's X^{2} statistic.
