Mathematics of probability (Q2841742)

The author presents an intermediate level (undergraduate/graduate) course on probability theory for students which are familiar with some basic facts on topology and are able to follow analytical reasoning (measure theory is not assumed and it is developed to the extent needed). Even if the matter is standard, the author points out some beautiful arguments in many proofs and considers interesting complementary topics. Selected exercises and notes at the end of each chapter allow the reader to practice reasoning and to go further.NEWLINENEWLINEThe first chapter explains basic concepts in finite and countable sample spaces. The random walk example is a starting point to develop De Moivre's theorem, the arcsine law, conditioning and independence, recurrence and transience. The case of uncountable sample spaces is described in the second chapter: the Lebesgue measure is obtained as an image of a symmetric Bernoulli measure. Then Lebesgue's theory of integration is described. As a complementary topic we find a connection between Gaussian computation and the Stirling gamma function formula.NEWLINENEWLINEChapter 3 concerns results for sums of independent variables and finishes with a proof of the strong law of large numbers based on Kolmogorov's inequality. The central limit theorem, Lindeberg's theorem and topics on Gaussian random variables (including the concentration phenomenon) are contained in Chapter 4. The most important topics on discrete Markov chains are described in Chapter 5, including the gambler's ruin problem, the Doeblin's condition, the existence of an invariant measure, the ergodic theorem.NEWLINENEWLINEChapter 6 is devoted to the study of continuous time Markov chains (transition probability functions and Chapman-Kolmogorov equation, Poisson process, Lévy's construction of the Brownian motion, elementary properties of the Brownian path, a discussion of the Ornstein-Uhlenbeck process). The final chapter concerns the theory of martingales (discrete and continuous). Several applications are given, for instance, the Radon-Nikodým theorem, exchangeable sequences and Hewitt-Savage 0-1 law, how to connect differential equations and probability by using continuous martingales, and in particular Brownian motion (Feynman-Kac formula).