This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Strong Law of Large Numbers Theorem. (Etemadi 1981) Let X 1 ,X 2 ,... be independent, identically distrib uted, integrable random variables, with common expected value μ . Let S n = ∑ n i =1 X i . Then S n /n → μ as n → ∞ almost surely. Proof. For each i ∈ N , define bounded random variables Y i = X i { X i  ≤ i } with means μ i = P Y i , and let T n = ∑ n i =1 Y i . (i) Claim: There is no loss of generality in assuming X i ≥ 0. Note that if X i satisfy the conditions of the Theorem, then so do X ± i ≥ 0: they are clearly iid, and integrable since X ± i ≤  X i  . So if the Theorem holds for nonnegative random variables, we deduce S n n = ∑ n i =1 X + i n + ∑ n i =1 X i n → P X + 1 P X 1 = μ as n → ∞ almost surely. (ii) a Claim: P T n /n → μ as n → ∞ .  P T n /n μ  ≤ 1 n n X i =1 P  X i { X i ≤ i }  X i  ≤ P X 1 1 n n X i =1 { X i > i } ≤ P X 1 ‡ 1 ∧ X 1 n · ....
View
Full Document
 Spring '09
 DavidPollard
 Law Of Large Numbers, Probability, Probability theory, Tn, Supremum, kN kN kN

Click to edit the document details