By Valeriy Skorokhod

The ebook is an creation to chance written through one in every of the famous specialists during this sector. Readers will find out about the fundamental thoughts of chance and its purposes, getting ready them for extra complicated and really expert works.

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**Extra info for Basic principles and applications of probability theory**

**Sample text**

Since it is a random variable and is determined up to sets of measure 0, we emphasize that all of the equalities (and inequalities) below are understood to hold with probability 1. 2 Deﬁnition of Probability Space 33 I. Conditional expectation is an additive function of random variables. This means the following. Let ξn = ξn (ω) be a ﬁnite sequence of random variables. Then E ξn |E = E(ξn |E) . 8). II. Let η be E-measurable and let ξ be such that E|ξη| < ∞ and E|ξ| < ∞. Then E(ξη | E) = ηE(ξ | E) .

1 Independent Algebras It will be recalled that algebras A1 , . . 1) i=1 for any choice of A1 ∈ A1 , . . , An ∈ An (see p. 24). (i) (i) Let Ai , i = 1, . . , l, be ﬁnite algebras and A1 , . . , Ani be the atoms of Ai . 1. Algebras Ai are independent if and only if for any ki ≤ ni , l (i) Ak i P i=1 l = (i) P(Aki ). 2) i=1 Proof. The necessity is obvious. It is easy to prove the suﬃciency by induction making use of the following trivial assertion: if A and B are independent, A and C are independent and B ∩C = ∅, then A and B ∪C are also independent.

Let ga (x) = −a for x < −a, ga (x) = x for |x| ≤ a and ga (x) = a for x > a. By Lebesgue’s dominated convergence theorem, limn→∞ Ega (ξn ) = Ega (ξ). By the uniform integrability, it follows that sup E|ga (ξn ) − ξn | ≤ sup E|ξn |I{|ξn |>a} , n n and this quantity may be made arbitrarily small by the choice of a. 2. Let us show that a may be chosen so that Eξn I{ξn >a} < ε for all n and any 38 2 Probability Space positive ε. To this end, it suﬃces that Efa (ξn ) < ε, where fa (x) = 0 for x < a/2, fa (x) = 2x − a for a/2 ≤ x ≤ a and fa (x) = x for x > a.