By Geiss C., Geiss S.
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Extra resources for An introduction to probability theory
Let H be the class of bounded F1 × F2 -measurable functions f : Ω1 × Ω2 → ❘ such that (a) the functions ω1 → f (ω1 , ω20 ) and ω2 → f (ω10 , ω2 ) are F1 -measurable and F2 -measurable, respectively, for all ωi0 ∈ Ωi , (b) the functions ω1 → Ω2 f (ω1 , ω2 )dP2 (ω2 ) and ω2 → Ω1 f (ω1 , ω2 )dP1 (ω1 ) are F1 -measurable and F2 -measurable, respectively, 54 CHAPTER 3. INTEGRATION (c) one has that f (ω1 , ω2 )d(P1 × P2 ) = Ω1 ×Ω2 Ω1 Ω2 Ω2 Ω1 = f (ω1 , ω2 )dP2 (ω2 ) dP1 (ω1 ) f (ω1 , ω2 )dP1 (ω1 ) dP2 (ω2 ).
S. Then f is integrable and one has that ❊f = lim ❊fn. n Proof. Applying Fatou’s Lemma gives ❊f = ❊ lim inf fn n→∞ ≤ lim inf ❊fn ≤ lim sup ❊fn ≤ n→∞ n→∞ ❊ lim sup fn = ❊f. n→∞ Finally, we state a useful formula for independent random variable. 8 If f and g are independent and ∞, then ❊|f g| < ∞ and ❊f g = ❊f ❊f. The proof is an exercise. 3 CHAPTER 3. INTEGRATION Connections to the Riemann-integral In two typical situations we formulate (without proof) how our expected value connects to the Riemann-integral.
8 . .. 2 Some applications We start with two fundamental examples of convergence in probability and almost sure convergence, the weak law of large numbers and the strong law of large numbers. 1 [Weak law of large numbers] Let (fn )∞ n=1 be a sequence of independent random variables with ❊f1 = m and ❊(f1 − m)2 = σ2. 2. SOME APPLICATIONS 65 Then f1 + · · · + fn P −→ m n that means, for each ε > 0, lim P n ω:| n → ∞, as f1 + · · · + fn − m| > ε n → 0. Proof. 9) we have that P f1 + · · · + fn − nm >ε n ω: ≤ ❊|f1 + · · · + fn − nm|2 = ❊( n 2 ε2 n k=1 (fk n 2 ε2 2 − m)) nσ 2 →0 n 2 ε2 = as n → ∞.