Idézet:This book can serve as a first course on measure theory and measure theoretic probability for upper undergraduate and graduate students of mathematics, statistics and probability. Starting from the basics, the measure theory part covers Caratheodory's theorem, Lebesgue-Stieltjes measures, integration theory, Fatou's lemma, dominated convergence theorem, basics of Lp spaces, transition and product measures, Fubini's theorem, construction of the Lebesgue measure in Rd, convergence of finite measures, Jordan-Hahn decomposition of signed measures, Radon-Nikodym theorem and the fundamental theorem of calculus.
The material on probability covers standard topics such as Borel-Cantelli lemmas, behaviour of sums of independent random variables, 0-1 laws, weak convergence of probability distributions, in particular via moments and cumulants, and the central limit theorem (via characteristic function, and also via cumulants), and ends with conditional expectation as a natural application of the Radon-Nikodym theorem. A unique feature is the discussion of the relation between moments and cumulants, leading to Isserlis' formula for moments of products of Gaussian variables and a proof of the central limit theorem avoiding the use of characteristic functions.

Idézet:A kódrészlet megtekintéséhez be kell jelentkezned, vagy nincs jogosultságod a tartalom megtekintéséhez.

Idézet:A kódrészlet megtekintéséhez be kell jelentkezned, vagy nincs jogosultságod a tartalom megtekintéséhez.