Distributional approximation and Stein's method
Development of Stein's method as a probabilistic tool: extension to new limit distributions; technical results concerning solutions of Stein equations; study of rates of convergence that are faster than the Berry-Esseen rate. Application of Stein's method to problems in mathematical statistics, random graph theory and queuing theory.
Useful online references for Stein's method are Yvik Swan's A gateway to Stein's Method and Ivan Nourdin's Malliavin-Stein webpage.
Quantitative limit theorems in mathematical statistics
Explicit errors bounds for distributional approximation in the multivariate delta method; the normal approximation of the posterior distribution in exponential families; the normal approximation of the maximum likelihood estimator; and the chi-square approximation of the Friedman, Pearson and power divergence family of statistics.
Probability distributions
Distributional properties of the variance-gamma, generalized hyperbolic and Conway-Maxwell-Poisson distributions, and the product of correlated normal random variables.
Networks
Poisson and compound Poisson approximation of the distribution of subgraph counts in stochastic block models and random graphs with multiple edges. Network comparison and development of new measures to assess network similarity.
Special functions
Mostly modified Bessel functions and related functions, particularly inequalities for modified Bessel, modified Struve and modified Lommel functions and their integrals. Applications of special functions to probability, particularly Stein's method and the study of probability distributions.