Multivariate Self-Normalized Processes with Matrix Normalization

The general framework of self-normalization in Chap. 10 and the method of mixtures in Chap. 11 has been extended by de la Peña et al. (2008) to the multivariate setting in which At is a vector and Bt is a positive definite matrix. Section 14.1 describes the basic concept of matrix square roots and the literature on its application to self-normalization. Section 14.2 extends the moment and exponential inequalities in Chap. 13 to multivariate self-normalized processes. Section 14.3 describes extensions of the boundary crossing probabilities in Sect. 11.3 and the law of the iterated logarithm in Sect. 13.3 to multivariate self-normalized processes with matrix normalization.