My research interests largely fall in to the area of probability theory including large sample theory, strong and weak limit theorems, convergence rates, and Bayesian modeling

  • Limit theorems for Banach space valued random elements
  • Limit theorems for multi-indexed sums and random fields
  • Statistical analysis and modeling of manifold valued data

Limit Theorems

In statistics, the three foundational limit laws from probability are the strong law of large numbers, the weak law of large numbers, and the central limit theorem. These results have been extended in many different directions such as for independent but not identically distributed random variables. One important extension came from stochastic process theory, where a stochastic process was viewed as a random element in a function space. This began the study of Hilbert and Banach space value random elements and their limit laws. In another direction was the study of multi-indexed random elements as well as random fields. The partially ordered structure of the multi-indexes led to the need for subtantially different proof techniques for establishing their limit laws. My research in this area is focued on finding conditions under which various limit laws hold in these settings.

Publish Papers

  • R. Parkerand A. Rosalsky, On complete convergence in mean for double sums ofindependent random elements in Banach spaces.Lobachevskii J. Math.(RussianAcademy of Sciences),38, 177-191 (2017)
  • R. Parkerand A. Rosalsky, Strong laws of large numbers for double sums ofBanach space valued random elements.Acta Mathematica Sinica, English Series,35, 583-596 (2019)
  • R. Parkerand A. Rosalsky, On almost certain convergence of double series ofrandom elements and the rate of convergence of tail series.Stochastics, 1-27(2020).

Manifold Valued Data

Manifold valued data has become very common. Examples include shape data, directions (unit vectors), symmetric positive definite matrices, orthogonal matrices, and many more. These types of data have arisen from many areas such as medical imaging, e.g. shapes of subcortical structures and how they change between healthy and diseased groups, vector cardiogram data, and movement ecology and meteorology which study directions. Since the data lie on a manifold, we lose the vector space structure that we use so often and standard statistical methods can fail. My interests in this area mainly lie in Bayesian modeling, in particular semi/non-parametric Bayesian modeling, of data lying on these different manifolds.