By Geiss C., Geiss S.

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**Additional info for An introduction to probability theory**

**Example text**

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 → ∞.

Proof. We simply have λP({ω : f (ω) ≥ λ}) = λ❊1I{f ≥λ} ≤ ❊f 1I{f ≥λ} ≤ ❊f. 2 [convexity] A function g : ❘ → ❘ is convex if and only if g(px + (1 − p)y) ≤ pg(x) + (1 − p)g(y) for all 0 ≤ p ≤ 1 and all x, y ∈ ❘. Every convex function g : ❘ → ❘ is (B(❘), B(❘))-measurable. 3 [Jensen’s inequality] If g : f : Ω → ❘ a random variable with ❊|f | < ∞, then ❘ → ❘ is convex and g(❊f ) ≤ ❊g(f ) where the expected value on the right-hand side might be infinity. 6. SOME INEQUALITIES 59 Proof. Let x0 = ❊f . Since g is convex we find a “supporting line”, that means a, b ∈ ❘ such that ax0 + b = g(x0 ) and ax + b ≤ g(x) for all x ∈ ❘.

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 → ∞. Using a stronger condition, we get easily more: the almost sure convergence instead of the convergence in probability. 2 [Strong law of large numbers] Let (fn )∞ n=1 be a sequence of independent random variables with ❊fk = 0, k = 1, 2, .