In this talk we present limit theorems for high dimensional data that is characterized by small sample sizes relative to the dimension of the data. In particular, we provide an infinite-dimensional framework to study statistical models that involve situations in which (i) the number of parameters increase with the sample size (that is allowed to be random) and (ii) there is a possibility of missing data. Under a variety of tail conditions on the components of the data, we provide conditions for the law of large numbers, as well as various results concerning the rate of convergence in these models. We also present central limit theorems in this setting, some which involve data driven coordinate-wise normalizations.
