F13

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14.381 F13 Recitation 1 notes: Modes of Convergence

Proof. (⇒) Let Ω0 = {ω : limn Xn(ω) = X(ω)}. Suppose P (Ω0) = 1. Let > 0 be given. →∞ Let Am = ∩k=m{|Xk − X| ≤ }. Then Am ⊂ Am+1 ∀m and limm P (Am) = P (∪m=1Am) by →∞ continuity of probability measure. For each ω0, there exists m(ω0) such that |Xk(ω0)−X(ω0)| ≤ for all k ≥ m(ω0). Therefore, ∀ω0 ∈ Ω0, ω0 ∈ Am for some m and we can conclude that Ω0 ⊂ ∪m=1Am and 1 = P (Ω0) ≤ P (∪m=1Am) = limm P (A ...

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ژورنال

عنوان ژورنال: Arthropod Management Tests

سال: 2012

ISSN: 2155-9856

DOI: 10.4182/amt.2012.f13