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Prove the Corollary to Theorem 1 in [P}§4.1]. Hint. Use the fact that the distribution function of the variable η takes only the values 0 and 1.

In the formal construction of a course in the theory of probability, limit theorems appear as a kind of superstructure over elementary chapters, in which all problems have finite, purely arithmetical character. In reality, however, the epistemological value of the theory of probability is revealed only by limit theorems. Moreover, without limit theorems it is impossible to understand the real content of the primary concept of all our sciences – the concept of probability.

The theory of probability, as a mathematical discipline, can and should be developed from axioms in exactly the same way as Geometry and Algebra. This means that after we have defined the elements to be studied and their basic relations, and have stated the axioms by which these relations are to be governed, all further exposition must be based exclusively on these axioms, independent of the usual concrete meaning of these elements and their relations.

In Chapter I (§12), for finite probability spaces, we took the basic idea to be that of Markov dependence between random variables. We also presented a variety of examples and considered the simplest regularities that are possessed by random variables that are connected by a Markov chain.

Let us consider an experiment of which all possible results are included in a finite number of outcomes ω 1,..., ω N . We do not need to know the nature of these outcomes, only that there are a finite number N of them.

We say that the function F = F(x), defined on ℝ m , is continuous at the pointx ∈ ℝ m if, for every e > 0, one can find a δ > 0, such that |F(x) − F(y)| < e for all y ∈ ℝ m that satisfy $$\begin{array}{rcl} x - \delta e < y < x + \delta e,& & \\ \end{array}$$ where \(e = (1,\ldots ,1) \in {\mathbb{R}}^{m}\). The sequence of distribution functions (F n ) n≥1 is said to converge in general to the distribution function F (notation: F n ⇒ F) if F n (x) → F(x) as n →∞, for any x ∈ ℝ m at which the function F = F(x) is continuous.

Let T be any measure preserving transformation acting on the sample space Ω and let ξ = ξ(ω), ω ∈ Ω, be any random variable, chosen so that the expected value Eξ(ω) exists. Prove that Eξ(ω) =Eξ(Tω).

According to the definition given in the preceding chapter, a random sequence ξ = (ξ1, ξ2, ...) is stationary in the strict sense if, for every set B ∈ ℛ(R ∞) and every n ≥ 1, $$P\left\{ {\left( {{\xi _1},{\xi _2},...} \right) \in B} \right\} = P\left\{ {\left( {{\xi _{n + 1}},{\xi _{n + 2}},...} \right) \in B} \right\}.$$ (1) It follows, in particular, that if \(\xi _1^2 < \infty \) then Eξ n is independent of n: $$E{\xi _n} = E{\xi _1},$$ (2) and the covariance cov(ξ n+m ξ n ) = E(ξ n+m − Eξ n+m )(ξ n − E ξ n ) depends only on m: $$\operatorname{cov} \left( {{\xi _{n + m}},{\zeta _n}} \right) = \operatorname{cov} \left( {{\xi _{1 + m}},{\zeta _1}} \right).$$ (3)