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ThmDex – An index of mathematical definitions, results, and conjectures.
P3207
Fix ε>0 and denote Zn,m:=Xn,mI{|Xn,m|λn}, SXn:=nm=1Xn,m, SZn:=nm=1Zn,m, and μn:=nm=1EZn,m. From result R4737: , we have the upper bound P(|SXnμnλn|>ε)P(SXnSZn)+P(|SZnμnλn|>ε) Notice that, by construction, we have {Xn,mZn,m}={|Xn,m|>λn} for all n1,2,3,. Applying results
(i) R4740:
(ii) R4738: Finite subadditivity of probability measure

as well as hypothesis (iv), we can estimate the first term on the right-hand side by P(SXnSZn)P(nm=1{Xn,mZn,m})P(nm=1{|Xn,m|>λn})nm=1P(|Xn,m|>λn)0 as n. Next, applying results
(i) R4741: Probabilistic Chebyshov's inequality for square function
(ii) R4687: Additivity of variance for a finite number of independent random real numbers
(iii) R4688: Second moment upper bound to real variance

as well as hypothesis (v), we have P(|SZnμnλn|>ε)1ε2E|SZnμnλn|2=1ε2λ2nVar(SZn)=1ε2λ2nnm=1Var(Zn,m)1ε2λ2nnm=1EZ2n,m0 as n. Combining these results, we find that P(|nm=1Xn,mE(Xn,mI{|Xn,m|λn})λn|>ε)=P(|nm=1Xn,mEZn,mλn|>ε)=P(|nm=1Xn,mnm=1EZn,mλn|>ε)=P(|SXnμnλn|>ε)P(SXnSZn)+P(|SZnμnλn|>ε)0 as n.