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(|SXn−μnλn|>ε)≤P(SXn≠SZn)+P(|SZn−μnλn|>ε)
Notice that, by construction, we have
{Xn,m≠Zn,m}={|Xn,m|>λn} for all
n∈1,2,3,…. Applying results
as well as hypothesis (iv), we can estimate the first term on the right-hand side by
P(SXn≠SZn)≤P(n⋃m=1{Xn,m≠Zn,m})≤P(n⋃m=1{|Xn,m|>λn})≤n∑m=1P(|Xn,m|>λn)→0
as
n→∞. Next, applying results
as well as hypothesis (v), we have
P(|SZn−μnλn|>ε)≤1ε2E|SZn−μnλn|2=1ε2λ2nVar(SZn)=1ε2λ2nn∑m=1Var(Zn,m)≤1ε2λ2nn∑m=1EZ2n,m→0
as
n→∞. Combining these results, we find that
P(|n∑m=1Xn,m−E(Xn,mI{|Xn,m|≤λn})λn|>ε)=P(|n∑m=1Xn,m−EZn,mλn|>ε)=P(|∑nm=1Xn,m−∑nm=1EZn,mλn|>ε)=P(|SXn−μnλn|>ε)≤P(SXn≠SZn)+P(|SZn−μnλn|>ε)→0
as
n→∞.
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