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# Etemadi - Strong Law of Large Numbers Theorem(Etemadi 1981...

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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 non-negative 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 · ....
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Etemadi - Strong Law of Large Numbers Theorem(Etemadi 1981...

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