Every probability course discusses the central limit theorem, which says that the sample mean of i.i.d. observations (under sufficiently weak conditions), after suitable centring and rescaling, converges in distribution to a standard normal random variable. But how can we quantify the rate of this distributional convergence?
Early attempts to answering this question relied on Fourier analysis, which was not entirely satisfactory at the conceptual level since it did not explain the special role of Gaussianity. This motivated Charles Stein to introduce an alternative approach based on the analysis of a probability-characterising operator, which has proven to be very powerful and flexible. Not only has this been generalised to more complicated statistical settings, it has also been used alongside other mathematical tools like Malliavin calculus and optimal transport to tackle different problems in modern probability theory.
This project will begin with an overview of the Stein philosophy, including the construction of Stein operator for the characterisation of probability distribution and the quantitative analysis of normal approximation. Several avenues of further investigations are then available, including (but not limited to):
Applications of Poisson approximation (Stein-Chen method)
Stein's method beyond the Gaussian/Poisson setting (e.g. stable approximations, etc.)
Stein's method of exchangeable pairs
Malliavin-Stein method and fourth moment theorem
Applications of Stein's method in probability problems
2H Probability and Analysis in Many Variables are essential.
There is a rich literature on Stein's method and one can have a look at the list compiled here: https://sites.google.com/site/steinsmethod/