Titlebook: Probability Theory; Alexandr A. Borovkov Textbook 2013 Springer-Verlag London Ltd., part of Springer Nature 2013 Ergodicity.Large Deviatio

SKIFF

An Arbitrary Space of Elementary Events,ecessarily countable. The concepts of algebra and sigma-algebra of sets are introduced and discussed in detail. Then the axioms of probability and, more generally, measure are presented and illustrated by several fundamental examples of measure spaces. The idea of extension of a measure is discussed

英寸

记忆法

allergy

Sequences of Independent Trials with Two Outcomes,nomial probabilities is proved in Sect. . using Stirling’s formula (covering both the normal approximation zone and the large deviations zone). The same section also contains a refinement of that result, including a bound for the relative error of the approximation, and an extension of the local lim

Fecundity

小步舞

Characteristic Functions,ated to the nature of the underlying distributions. Section . presents the proofs of the inversion formulas for both densities and distribution functions, and also in the space of square integrable functions. Then the fundamental continuity theorem relating pointwise convergence of characteristic fu

Ovulation

Sequences of Independent Random Variables. Limit Theorems,ntically distributed summands, both using the apparatus of characteristic functions. Section . establishes general conditions for the Weak Law of Large Numbers for general sequences of independent random variables and also conditions for the respective convergence in mean. Section . presents the Cen

outer-ear

Large Deviation Probabilities for Sums of Independent Random Variables,n probabilities, Cramér’s condition is introduced and the main properties of the Cramér and Laplace transforms are discussed in Sect. .. A separate subsection is devoted to an in-depth analysis of the key properties of the large deviation rate function, followed by Sect. . establishing the fundament

横截,横断

Renewal Processes,ishes the basic terminology and proves the integral renewal theorem in the case of non-identically distributed random variables. The classical Key Renewal Theorem in the arithmetic case is proved in Sect. ., including its extension to the case where random variables can assume negative values. The l

责难

Properties of the Trajectories of Random Walks. Zero-One Laws,e concepts of lower and upper functions are introduced. Section . contains the first Kolmogorov inequality and several theorems on convergence of random series. Section . presents Kolmogorov’s Strong Law of Large Numbers and Wald’s identity for stopping times. Sections . and . are devoted to the Str