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

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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 . Let S n = n i =1 X i . Then S n /n as n almost surely. Proof. For each i N , define bounded random variables Y i = X i {| X i | i } with means 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 = as n almost surely. (ii) a Claim: P T n /n as n . | P T n /n- | 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...

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