Theory and applications of multivariate self-normalized processes

Multivariate self-normalized processes, for which self-normalization consists of multiplying by the inverse of a positive definite matrix (instead of dividing by a positive random variable as in the scalar case), are ubiquitous in statistical applications. In this paper we make use of a technique called “pseudo-maximization” to derive exponential and moment inequalities, and bounds for boundary crossing probabilities, for these processes. In addition, Strassen-type laws of the iterated logarithm are developed for multivariate martingales, self-normalized by their quadratic or predictable variations.