Probability theory. Introductory course with applications to mathematical statistics (Q2911451)

This textbook originates from a number of lectures given by the author at EPF Lausanne in French for students of Physics during their third semester.NEWLINENEWLINE The text is divided in 16 chapters of rather similar size, each of which roughly corresponds to the teaching content of a week (maybe with the exception of the longer chapter on the central limit theorem). Each of the chapters ends with a section of exercises.NEWLINENEWLINEAfter a compact reminder of analytic and linear-algebraic background and a short introduction about the historic foundations of probability, the course on probability theory starts with the formulation of Kolmogorov's axioms. Already in this (the second) chapter, the author introduces an elementary notion of a Gibb's measure and the Ising model.NEWLINENEWLINEThe first main part of the book, which, in the reviewer's opinion, could be labeled ``elementary probability theory'' touches the classical subjects. The author starts with ``des boules et des boîtes'', that is with combinatorics, followed by the introduction of the (abstract) concepts of conditional probability and independence. Interestingly, it is only then that the author introduces particular probability spaces on the real line and in a Euclidean space. In a swift sequence of three chapters, he introduces the concept of a random variable and a random vector, of expectation and variance and the classical inequalities associated with Markov, Chebyshev and Hoeffending. The last inequality is applied to introduce an elementary version of the large deviation principle which is applied in an example to the Ising model.NEWLINENEWLINEA second part elaborates classical limit theorems on the basis of the previous chapters. In the ninth chapter, the reader gets acquainted to the notion of a Markov chain on a finite state space and the ergodic theorem for this class of processes is shown. This is followed by a chapter on the law of large numbers. First, the author proves the weak version and applies this to weakly correlated processes. He shows the strong version and gives as three major applications the Glivenko-Cantelli lemma, a sketch of the Monte Carlo method and the convergence of the sojourn times for ergodic Markov chains. Chapter 11 is a short introduction of the random walk and starts with a sketch of the law of the iterated logarithm. Main properties obtained are the reflection principle, a discrete version of the arcsine-law of the sojourn times, as well as recurrence and transience. The part on probability theory finishes with a chapter on the cental limit theorem.NEWLINENEWLINEIn a preparatory chapter, the author gives the heuristics for the renormalized convergence of the random walk on \(\mathbb{Z}\) to a Gaussian. In a second subchapter, he shows the theorem of De Moivre-Laplace followed by the central limit theorem including the Berry-Esseen bound. In fact, the author presents a proof under Lindeberg conditions. Interestingly, only after these results, the author introduces the notion of weak convergence and reinterprets the previous results in this light.NEWLINENEWLINEThe third part is devoted to an introduction to basic statistic concepts, as point estimates, the least square method, interval estimates and tests. NEWLINENEWLINEThe text ends with a section containing solutions to selected exercises.NEWLINENEWLINEApart from the chapter on the central limit theorem which seems slightly too technical for third-semester students, the text is written in a very gently readable, clear and pedagogic style. It convinces, in particular, with many interesting examples from physics/statistical mechanics. To sum up, this textbook yields an excellent introduction to the main results of probability theory and statistics with an emphasis on examples in physics for students with a good analytical background.