Sequences of Independent Random Variables. Limit Theorems

The chapter opens with proofs of Khintchin’s (weak) Law of Large Numbers (Sect. 8.1) and the Central Limit Theorem (Sect. 8.2) the case of independent identically distributed summands, both using the apparatus of characteristic functions. Section 8.3 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 8.4 presents the Central Limit Theorem in the triangular array scheme (the Lindeberg–Feller theorem) and its corollaries, illustrated by several insightful examples. After that, in Sect. 8.5 an alternative method of compositions is introduced and used to prove the Central Limit Theorem in the same situation, establishing an upper bound for the convergence rate for the uniform distance between the distribution functions in the case of finite third moments. This is followed by an extension of the above results to the multivariate case in Sect. 8.6. Section 8.7 presents important material not to be found in other textbooks: the so-called integro-local limit theorems on convergence to the normal distribution (the Stone–Shepp and Gnedenko theorems), including versions for sums of random variables depending on a parameter. These results will be of crucial importance in Chap. 9, when proving theorems on exact asymptotic behaviour of large deviation probabilities. The chapter concludes with Sect. 8.8 establishing integral, integro-local and local theorems on convergence of the distributions of scaled sums on independent identically distributed random variables to non-normal stable laws.

1. 1.

   The second version was communicated to us by A.I. Sakhanenko.

2. 2.

   There exists an alternative “direct” proof of Theorem 8.3.1 using not ch.f.s but the so-called truncated random variables and estimates of their variances. However, because of what follows, it is more convenient for us to use here the machinery of ch.f.s.

3. 3.

   There exist several conditions characterising admissible dependence of ξ _k,n . Such considerations are beyond the scope of the present book, but can be found in the special literature. See e.g. [20].

4. 4.

   If η⊂=Φ _0,1 then \(c_{3} =\mathbf{E}|\eta|^{3} =\frac{2}{\sqrt{2\pi}} \int _{0}^{\infty} x^{3} e^{-{x^{2}}/2}dx = \frac{4}{\sqrt{2\pi}} \int_{0}^{\infty} te^{-t}dt =\frac{4}{\sqrt{2\pi}}\).

5. 5.

   See [33].

Authors and Affiliations

1. Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, Russia

   Alexandr A. Borovkov

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