In this section, we will study the strong convergence for a class of random variables satisfying the Rosenthal-type maximal inequality by using a different method from that of Sung [8]. As an application, the Marcinkiewicz-Zygmund-type strong law of large numbers is obtained.
Proof.
We only need to prove that (3) holds for Yni=-bnI(aniXi<-bn)+aniXiI(|aniXi|≤bn)+bnI(aniXi>bn). The proof for Yni=aniXiI(|aniXi|≤bn) is analogous.
Without loss of generality, we may assume that ∑i=1n|ani|α≤n. It is easy to check that for any ɛ>0,
(7)(max1≤j≤n|∑i=1janiXi|>ɛbn) ⊂(max1≤i≤n|aniXi|>bn)⋃(max1≤j≤n|∑i=1jYni|>ɛbn),
which implies that
(8)P(max1≤j≤n|∑i=1janiXi|>ɛbn) ≤P(max1≤i≤n|aniXi|>bn)+P(max1≤j≤n|∑i=1jYni|>ɛbn) ≤∑i=1nP(|aniXi|>bn) +P(max1≤j≤n|∑i=1j(Yni-EYni)|>ɛbn-max1≤j≤n|∑i=1jEYni|).
Firstly, we will show that
(9)bn-1max1≤j≤n|∑i=1jEYni|→0, as n→∞.
When 1<α≤2, we have by EXn=0, Markov’s inequality and (6) that
(10)bn-1max1≤j≤n|∑i=1jEYni|≤∑i=1nP(|aniXi|>bn)+bn-1max1≤j≤n|∑i=1janiEXiI(|aniXi|>bn)|≤∑i=1nP(|aniX1|>bn)+bn-1∑i=1nE|aniX1|I(|aniX1|>bn)≤bn-α∑i=1n|ani|αE|X1|α+bn-α∑i=1n|ani|αE|X1|α≤2E|X1|α(logn)-α/γ→0, as n→∞.
When 0<α≤1, we have by Markov’s inequality and (6) again that
(11)bn-1max1≤j≤n|∑i=1jEYni| ≤∑i=1nP(|aniXi|>bn)+bn-1∑i=1nE|aniXi|I(|aniXi|≤bn) =∑i=1nP(|aniX1|>bn)+bn-1∑i=1nE|aniX1|I(|aniX1|≤bn) ≤bn-α∑i=1n|ani|αE|X1|α+bn-α∑i=1nE|aniX1|αI(|aniX1|≤bn) ≤bn-αnE|X1|α+bn-αnE|X1|α =2E|X1|α(logn)-α/γ→0, as n→∞.
By (10) and (11), we can get (9) immediately. Hence, for n large enough,
(12)P(max1≤j≤n|∑i=1janiXi|>ɛbn) ≤∑i=1nP(|aniXi|>bn)+P(max1≤j≤n|∑i=1j(Yni-EYni)|>ɛ2bn).
To prove (3), we only need to show that
(13)I≐∑n=1∞n-1∑i=1nP(|aniXi|>bn)<∞,(14)J≐∑n=1∞n-1P(max1≤j≤n|∑i=1j(Yni-EYni)|>ɛ2bn)<∞.
Firstly, we will prove (13). By ∑i=1n|ani|α≤n and (6), we can get that
(15)I≤∑n=1∞n-1bn-α∑i=1nE|aniX1|αI(|aniX1|>bn)≤∑n=1∞n-2(logn)-α/γ×∑i=1nE|aniX1|αI(∑i=1n|aniX1|α>n(logn)α/γ)≤∑n=1∞n-2(logn)-α/γ×∑i=1nE|aniX1|αI(|X1|α>(logn)α/γ)≤∑n=1∞n-1(logn)-α/γE|X1|αI(|X1|γ>logn)=∑n=1∞n-1(logn)-α/γ×∑m=n∞E|X1|αI(logm<|X1|γ≤log(m+1))=∑m=1∞E|X1|αI(logm<|X1|γ≤log(m+1))×∑n=1mn-1(logn)-α/γ≤{C∑m=1∞E|X1|αI(logm<|X1|γ≤log(m+1)), for α>γ,C∑m=1∞E|X1|αI(logm<|X1|γ≤log(m+1)) ×log logm, for α=γ,C∑m=1∞E|X1|αI(logm<|X1|γ≤log(m+1)) ×(logm)1-α/γ, for α<γ≤{CE|X1|α,for α>γ,CE|X1|αlog|X1|,for α=γ,CE|X1|γ,for α<γ<∞,
which implies (13).
In the following, we will prove (14). Let q>max{2,α,γ,(2γ/α)}. By Markov’s inequality and condition (4), we have
(16)J≤C∑n=1∞n-1bn-qE(max1≤j≤n|∑i=1j(Yni-EYni)|q)≤C∑n=1∞n-1bn-q∑i=1nE|Yni|q+C∑n=1∞n-1bn-q(∑i=1nEYni2)q/2≐J1+J2.
To prove (14), it suffices to show that J1<∞ and J2<∞.
For j≥1 and n≥2, denote
(17)Inj={1≤i≤n:nj+1<|ani|α≤nj}.
In view of ∑i=1n|ani|α≤n, it is easy to see that {Inj,j≥1} are disjoint and ⋃j=1∞Inj={1≤i≤n:ani≠0}. Hence, we have for all m≥1 that
(18)n≥∑i=1n|ani|α=∑{1≤i≤n:ani≠0}|ani|α=∑j=1∞ ∑i∈Inj|ani|α≥n∑j=1∞(j+1)-1♯Inj≥n∑j=m∞(j+1)-q/α(j+1)q/α-1♯Inj≥n∑j=m∞(j+1)-q/α(m+1)q/α-1♯Inj,
which implies that for all m≥1,
(19)∑j=m∞(j+1)-q/α♯Inj≤Cm1-q/α, n≥2.
By Cr’s inequality, (13) and (17), we can get that
(20)J1≤C∑n=1∞n-1∑i=1nP(|aniXi|>bn)+C∑n=1∞n-1bn-q∑i=1nE|aniXi|qI(|aniXi|≤bn)≤C∑n=2∞n-1bn-q∑i=1nE|aniX1|qI(|aniX1|≤bn)≤C∑n=2∞n-1-q/α(logn)-q/γ×∑j=1∞ ∑i∈InjE|aniX1|qI(|X1|≤(j+1)1/α(logn)1/γ)≤C∑n=2∞n-1-q/α(logn)-q/γ×∑j=1∞nq/αj-q/αE|X1|qI(|X1|≤(j+1)1/α(logn)1/γ)♯Inj≤C∑n=2∞n-1(logn)-q/γ×∑j=1∞j-q/αE|X1|qI(|X1|≤(logn)1/γ)♯Inj+C∑n=2∞n-1(logn)-q/γ×∑j=1∞j-q/α∑k=1jE|X1|qI(k1/α(logn)1/γ<|X1| ≤(k+1)1/α(logn)1/γ)♯Inj≐J11+J12.
If α>γ, we have by (19) and E|X1|α<∞ that
(21)J11≤C∑n=2∞n-1(logn)-q/γE|X1|qI(|X1|≤(logn)1/γ)≤C∑n=2∞n-1(logn)-α/γE|X1|αI(|X1|≤(logn)1/γ)≤C∑n=2∞n-1(logn)-α/γ<∞.
If α≤γ, we have by (6) and (19) that
(22)J11≤C∑n=2∞n-1(logn)-q/γE|X1|qI(|X1|≤(logn)1/γ)≤C∑n=2∞n-1(logn)-q/γ×∑m=2nE|X1|qI((log(m-1))1/γ<|X1|≤(logm)1/γ)=C∑m=2∞E|X1|qI((log(m-1))1/γ<|X1|≤(logm)1/γ)×∑n=m∞n-1(logn)-q/γ≤C∑m=2∞(logm)1-q/γE|X1|qI ×((log(m-1))1/γ<|X1|≤(logm)1/γ)≤C∑m=2∞E|X1|γI((log(m-1))1/γ<|X1|≤(logm)1/γ)≤CE|X1|γ<∞.
By (21) and (22), we can get that J11<∞. Next, we will prove that J12<∞.
It follows by (6) and (19) again that
(23)J12=C∑n=2∞n-1(logn)-q/γ×∑k=1∞E|X1|qI(k1/α(logn)1/γ<|X1| ≤(k+1)1/α(logn)1/γ)×∑j=k∞j-q/α♯Inj≤C∑n=2∞n-1(logn)-q/γ×∑k=1∞k1-q/αE|X1|qI ×(k1/α(logn)1/γ<|X1|≤(k+1)1/α(logn)1/γ)≤C∑n=2∞n-1(logn)-α/γ×∑k=1∞E|X1|αI(k1/α(logn)1/γ<|X1| ≤(k+1)1/α(logn)1/γ)=C∑n=2∞n-1(logn)-α/γE|X1|αI(|X1|>(logn)1/γ)=C∑n=2∞n-1(logn)-α/γ×∑m=n∞E|X1|αI((logm)1/γ<|X1|≤(log(m+1))1/γ)=C∑m=2∞E|X1|αI((logm)1/γ<|X1|≤(log(m+1))1/γ)×∑n=2mn-1(logn)-α/γ≤{C∑m=1∞E|X1|αI(logm<|X1|γ≤log(m+1)), for α>γ,C∑m=1∞E|X1|αI(logm<|X1|γ≤log(m+1)) ×log logm, for α=γ,C∑m=1∞E|X1|αI(logm<|X1|γ≤log(m+1)) ×(logm)1-α/γ, for α<γ≤{CE|X1|α,for α>γ,CE|X1|αlog|X1|,for α=γ,CE|X1|γ,for α<γ<∞.
By J11<∞ and J12<∞, we can get that J1<∞.
To prove (14), it suffices to show that J2<∞. By Cr’s inequality, conditions (5) and (6), we can get that
(24)J2≤C∑n=1∞n-1[∑i=1nP(|aniXi|>bn)]q/2+C∑n=1∞n-1bn-q[∑i=1nE|aniXi|2I(|aniXi|≤bn)]q/2≤C∑n=1∞n-1[∑i=1nbn-αE|aniX1|αI(|aniX1|≤bn)]q/2≤C(E|X1|α)q/2∑n=1∞n-1(logn)-αq/2γ<∞.
Therefore, (14) follows from (16) and J1<∞, J2<∞ immediately. This completes the proof of the theorem.
The following result provides the Marcinkiewicz-Zygmund-type strong law of large numbers for weighted sums ∑i=1naiXi of a class of random variables satisfying the Rosenthal-type maximal inequality.
If the Rosenthal type inequality for the maximal partial sum is replaced by the partial sum, then we can get the following complete convergence result for a class of random variables. The proof is similar to that of Theorem 2. So the details are omitted.