Lecture Notes on Multivariate Statistics

These are my lecture notes for the course Multivariate Analysis (MATH444) at EPFL. They have been developed and refined over three years of teaching the course and reflect both its structure and its emphasis as presented in class. They are not intended to serve as a comprehensive reference on multivariate statistics. Rather, their purpose is to provide a concise, coherent, and self-contained account of the material covered in the course. Throughout, proofs and examples have been selected with pedagogy in mind rather than maximal generality. As with any course, choices had to be made. Certain classical topics in multivariate statistics are therefore absent (e.g. discriminant analysis, factor analysis, clustering,...). The omissions reflect the limited time available in a one-semester course and my preference to focus on a smaller collection of ideas in greater depth.The intended audience is Master's-level students in mathematics and statistics. The course assumes familiarity with linear algebra, mathematical analysis, and the fundamentals of probability and statistics. The exposition aims to balance mathematical rigour with intuition and statistical relevance. I wrote these notes primarily as a teaching resource rather than as a scholarly publication. The presentation draws on a broad range of standard references, accumulated over the years, and no attempt has been made to systematically trace the provenance of every argument, example, or formulation. Some material appears essentially as found in the literature; other parts have been adapted, reorganised, or substantially reworked to reflect my preferred way of presenting the subject. In some cases, the selection, synthesis, and exposition of the material may be original, although similar treatments may well exist elsewhere. Over the course of three years, several corrections and improvements have resulted from the careful reading and feedback of students. Nevertheless, the notes undoubtedly still contain inaccuracies, omissions, typographical errors, and other bugs. Comments, corrections, and suggestions are therefore always welcome.