A Gut

A Spătaru

H Lanzinger

Jiaming Yi

LE Baum

M Katz

M Ledoux

P Chen

Pingyan Chen

PL Hsu

SH Sung

Soo Hak Sung

VV Petrov

W Li

Journal of Inequalities and Applications

Crossref

Chen et al. Journal of Inequalities and Applications  (2015) 2015:414 DOI 10.1186/s13660-015-0939-2RESEARCH Open AccessAn extension of the Baum-Katz theoremto i.i.d. random variables with generalmoment conditionsPingyan Chen1, Jiaming Yi2 and Soo Hak Sung3**Correspondence:sungsh@pcu.ac.kr3Department of AppliedMathematics, Pai Chai University,Taejon, 302-735, South KoreaFull list of author information isavailable at the end of the articleAbstractFor a sequence of i.i.d. random variables {X ,Xn ,n≥ 1} and a sequence of positive realnumbers {an,n≥ 1} with 0 < an/n1/p ↑ for some 0 < p < 2, the Baum-Katz completeconvergence theorem is extended to the {X ,Xn ,n≥ 1} with the general momentcondition∑∞n=1 nr–1P{|X| > an} <∞, where r ≥ 1. The relationship between thecomplete convergence and the strong law of large numbers is established.MSC: 60F15Keywords: Baum-Katz theorem; complete convergence; general momentcondition; strong law of large numbers1 Introduction andmain resultThe concept of complete convergence was ﬁrst introduced by Hsu and Robbins [] andhas played a very important role in probability theory. A sequence of random variables{Un,n ≥ } is said to converge completely to a constant C if ∑∞n= P{|Un – C| > ε} < ∞ forany ε > . Hsu and Robbins [] proved that the sequence of arithmetic means of inde-pendent and identically distributed (i.i.d.) random variables converges completely to theexpected value if the variance of the summands is ﬁnite. Their result has been generalizedand extended by many authors.The following result is well known.Theorem A Let r ≥  and  < p < . Let {X,Xn,n≥ } be a sequence of i.i.d. random vari-ables with partial sums Sn =∑nk=Xk , n≥ . Then the following statements are equivalent:E|X|rp <∞, (.)∞∑n=nr–P{|Sn – nb| > εn/p}<∞ ∀ε > , (.)∞∑n=nr–P{max≤m≤n |Sm –mb| > εn/p}<∞ ∀ε > , (.)where b =  if  < rp <  and b = EX if rp≥ .© 2015 Chen et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.Chen et al. Journal of Inequalities and Applications  (2015) 2015:414 Page 2 of 9When r = , each of (.)∼(.) is equivalent toSn – nbn/p →  a.s. (.)When r = , the equivalence of (.) and (.) is known as the Marcinkiewicz and Zyg-mund strong law of large numbers. Katz [] proved the equivalence of (.) and (.) forthe case of p = . Baum and Katz [] proved the equivalence of (.) and (.) for the caseof  < p < . The result of Baum and Katz was generalized and extended in several direc-tions. Some versions of the Baum and Katz theorem under higher-order moment condi-tions were established by Lanzinger [], Gut and Stadtmüller [], and Chen and Sung [].When p = , ≤ r < , and {Xn,n≥ } is a sequence of pairwise independent, but not nec-essarily identically distributed, random variables, Spătaru [] gave suﬃcient conditions for(.).It is interesting to ﬁndmore general moment conditions such that the complete conver-gence holds. In fact, Li et al. [] and Sung [] have done something. In particular, it is worthpointing out that Sung [] obtained the following complete convergence for pairwise i.i.d.random variables {X,Xn,n≥ }:∞∑n=n–P{∣∣∣∣∣n∑k=Xk – nEXI(|X| ≤ an)∣∣∣∣∣> εan}<∞ ∀ε > ,provided that∑∞n= P{|X| > an} <∞, where  < an/n ↑.Motivated by the work of Sung [], the aim of this paper is to obtain the complete con-vergence undermore general moment conditions. Ourmain result includes the Baum andKatz [] complete convergence and the Marcinkiewicz and Zygmund strong law of largenumbers.Now we state the main result. Some lemmas and the proof of the main result will bedetailed in next section.Theorem . Let r ≥  and  < p < . Let {X,Xn,n ≥ } be a sequence of i.i.d. randomvariables with partial sums Sn =∑nk=Xk , n≥ , and {an,n≥ } a sequence of positive realnumbers with  < an/n/p ↑. Then the following statements are equivalent:∞∑n=nr–P{|X| > an}<∞, (.)∞∑n=nr–P{|Sn – nbn| > εan}<∞ ∀ε > , (.)∞∑n=nr–P{max≤m≤n |Sm –mbn| > εan}<∞ ∀ε > , (.)where bn =  if  < p <  and bn = EXI(|X| ≤ an) if ≤ p < .When r = , each of (.)-(.) is equivalent toa–n (Sn – nbn)→  a.s. (.)Chen et al. Journal of Inequalities and Applications  (2015) 2015:414 Page 3 of 9Remark . When an = n/p for n ≥ , (.) is equivalent to (.). In this case, a–n ·nEXI(|X| > an) →  if  ≤ p <  and (.) holds. Hence, (.) ⇒ (.), (.) ⇒ (.), and(.) ⇒ (.) (in this case, r = ) follow from Theorem .. Although the converses do notfollow directly from Theorem ., the proofs can be done easily. When an = n/p(lnn)α forn≥ , where α > , (.) is equivalent to E|X|rp/(ln(|X| + ))αrp <∞.Throughout this paper, the symbol C denotes a positive constant that is not necessarilythe same one in each appearance, and I(A) denotes the indicator function of an event A.2 Lemmas and proofsTo prove the main result, the following lemmas are needed. Lemma . is the Rosenthalinequality for the sum of independent random variables; see, for example, Petrov [].Lemma . Let {Yn,n ≥ } be a sequence of independent random variables with EYn = and E|Yn|s < ∞ for some s ≥  and all n ≥ . Then there exists a positive constant Cdepending only on s such that for all n≥ ,E max≤m≤n∣∣∣∣∣m∑k=Yk∣∣∣∣∣s≤ C{ n∑k=E|Yk|s +( n∑k=EY k)s/}.Lemma . Under the assumptions of Theorem ., if  < p <  and (.) holds, thena–n · nEXI(|X| ≤ an) → as n→ ∞.Proof Since  < p < , by  < an/n/p ↑ we have  < an/n ↑ ∞. By (.) we have∞∑n=P{|X| > an}<∞.Therefore, by Lemma . in Sung [] we have the desired result. Lemma . Under the assumptions of Theorem ., if rp≥  and (.) holds, thena–n · nE|X|I(|X| ≤ an) ≤ Cn–/p.Proof By  < an/n/p ↑ we have ak/an ≤ (k/n)/p for any ≤ k ≤ n. Hence,a–n · nE|X|I(|X| ≤ an)= a–n · nn∑k=E|X|I(ak– < |X| ≤ ak)(set a = )≤ a–n · nn∑k=akP{ak– < |X| ≤ ak}≤ n–/pn∑k=k/pP{ak– < |X| ≤ ak}Chen et al. Journal of Inequalities and Applications  (2015) 2015:414 Page 4 of 9≤ n–/pn∑k=krP{ak– < |X| ≤ ak}(by rp≥ )≤ n–/p∞∑k=[(k + )r – kr]P{|X| > ak}≤ n–/p ·( + rr–∞∑k=kr–P{|X| > ak}).Set C =  + rr–∑∞k= kr–P{|X| > ak}. By (.), C <∞. So we complete the proof. Lemma . Under the assumptions of Theorem ., if s > rp and (.) holds, then∞∑n=nr– · a–sn nE|X|sI(|X| ≤ an)<∞.Proof By  < an/n/p ↑ we have ak/an ≤ (k/n)/p for any n≥ k. Hence,∞∑n=nr– · a–sn nE|X|sI(|X| ≤ an)=∞∑n=nr–a–snn∑k=E|X|sI(ak– < |X| ≤ ak)≤∞∑n=nr–a–snn∑k=askP{ak– < |X| ≤ ak}=∞∑k=askP{ak– < |X| ≤ ak} ∞∑n=knr–a–sn≤∞∑k=ks/pP{ak– < |X| ≤ ak} ∞∑n=knr––s/p≤ C∞∑k=krP{ak– < |X| ≤ ak}<∞.Therefore, the proof is completed. Lemma . Let {X,Xn ≥ } be a sequence of i.i.d. symmetric random variables, and{an,n≥ } a sequence of real numbers with  < an ↑ ∞. Suppose that∞∑n=n–P{∣∣∣∣∣n∑k=Xk∣∣∣∣∣> εan}<∞ ∀ε > . (.)Thena–nn∑k=Xk →  in probability. (.)Proof Set Sn =∑nk=Xk , n≥ . Note that for all ε > ,P{|Sn+| > εan+} ≤ P{|Sn| > εan+/}+ P{|Xn+| > εan+/}≤ P{|Sn| > εan/}+ P{|X| > εan+/}Chen et al. Journal of Inequalities and Applications  (2015) 2015:414 Page 5 of 9and P{|X| > εan+/} →  as n→ ∞. Hence, to prove (.), it suﬃces to prove thata–nSn →  in probability. (.)We will prove (.) by contradiction. Suppose that there exist a constant ε >  and a se-quence of integers {ni, i≥ } with ni ↑ ∞ such thatP{|Sni | > εani} ≥ ε for all i≥ .Without loss of generality, we can assume that ni < ni+. By the Lévy inequality (see, forexample, formula (.) in Ledoux and Talagrand []) we have∞∑n=n–P{|Sn| > εan/} ≥ ∞∑n=n–P{max≤k≤n|Sk| > εan/}≥ ∞∑i=ni∑n=ni+n–P{max≤k≤n|Sk| > εan/}≥ ∞∑i=ni∑n=ni+n–P{max≤k≤ni|Sk| > εani/}≥ ∞∑i=ni∑n=ni+n–P{|Sni | > εani/}= ∞∑i=ni∑n=ni+n–(P{|Sni | > εani/}+ P{|Sni – Sni | > εani/})≥ ∞∑i=ni∑n=ni+n–P{|Sni | > εani}≥ ε∞∑i=ni∑n=ni+n– =∞,which leads a contradiction to (.). Hence, (.) holds, and so the proof is completed. Proof of Theorem . We ﬁrst prove that (.) implies (.). By Lemma ., to prove (.),it suﬃces to prove that∞∑n=nr–P{max≤m≤n∣∣∣∣∣m∑k=(Xk – EXkI(|Xk| ≤ an))∣∣∣∣∣> εan}<∞ ∀ε > . (.)Note that{max≤m≤n∣∣∣∣∣m∑k=(Xk – EXkI(|Xk| ≤ an))∣∣∣∣∣> εan}⊂n⋃k={|Xk| > an} ∪{max≤m≤n∣∣∣∣∣m∑k=(XkI(|Xk| ≤ an)– EXkI(|Xk| ≤ an))∣∣∣∣∣> εan}.Chen et al. Journal of Inequalities and Applications  (2015) 2015:414 Page 6 of 9Hence, by (.), to prove (.) it suﬃces to prove that for all ε > ,∞∑n=nr–P{max≤m≤n∣∣∣∣∣m∑k=(XkI(|Xk| ≤ an)– EXkI(|Xk| ≤ an))∣∣∣∣∣> εan}<∞. (.)For any s≥ , by the Markov inequality and Lemma .,∞∑n=nr–P{max≤m≤n∣∣∣∣∣m∑k=(XkI(|Xk| ≤ an)– EXkI(|Xk| ≤ an))∣∣∣∣∣> εan}≤ C∞∑n=nr–a–sn E max≤m≤n∣∣∣∣∣m∑k=(XkI(|Xk| ≤ an)– EXkI(|Xk| ≤ an))∣∣∣∣∣s≤ C( ∞∑n=nr–{a–n nEXI(|X| ≤ an)}s/ +∞∑n=nr–a–sn E|X|sI(|X| ≤ an))= C(I + I).If rp≥ , taking s large enough such that r –  – s/p + s/ < –, by Lemma . we haveI ≤ C∞∑n=nr––s/p+s/ <∞.Since s > rp, I < ∞ by Lemma .. If  < rp < , taking s =  (in this case I = I), we haveI = I <∞ by Lemma . again. Hence, (.) holds for all ε > .It is trivial that (.) implies (.). Nowwe prove that (.) implies (.). Let {X ′,X ′n,n≥ }be an independent copy of {X,Xn,n≥ }. Then we also have∞∑n=nr–P{∣∣∣∣∣n∑k=X ′k – nbn∣∣∣∣∣> εan}<∞ ∀ε > .Hence,∞∑n=nr–P{∣∣∣∣∣n∑k=(Xk –X ′k)∣∣∣∣∣> εan}<∞ ∀ε > ,from which it follows that∞∑n=n–P{∣∣∣∣∣n∑k=(Xk –X ′k)∣∣∣∣∣> εan}<∞ ∀ε > .Then, by Lemma .,a–nn∑k=(Xk –X ′k) →  in probability.Chen et al. Journal of Inequalities and Applications  (2015) 2015:414 Page 7 of 9By the Lévy inequality (see, for example, formula (.) in Ledoux and Talagrand []), forany ﬁxed ε > ,P{max≤k≤n∣∣Xk –X ′k∣∣ > εan}≤ P{∣∣∣∣∣n∑k=(Xk –X ′k)∣∣∣∣∣> εan}→ as n→ ∞. Then, for all n large enough,P{max≤k≤n∣∣Xk –X ′k∣∣ > εan}≤ /.Therefore, by Lemma . in Ledoux and Talagrand [] and the Lévy inequality (see for-mula (.) in Ledoux and Talagrand []) we have that for all n large enough,nP{∣∣X –X ′∣∣ > εan}=n∑k=P{∣∣Xk –X ′k∣∣ > εan}≤ P{max≤k≤n∣∣Xk –X ′k∣∣ > εan}≤ P{∣∣∣∣∣n∑k=(Xk –X ′k)∣∣∣∣∣> εan}.Therefore,∞∑n=nr–P{∣∣X –X ′∣∣ > εan}<∞ ∀ε > . (.)Since P{|X| > an/} →  as n → ∞, |med(X)/(an/)| ≤  for all n large enough. By theweak symmetrization inequality we have that for all n large enough,P{|X| > an} ≤ P{∣∣X – med(X)∣∣ > an/} ≤ P{∣∣X –X ′∣∣ > an/}, (.)which, together with (.), implies that (.) holds.Finally, we prove that (.) and (.) are equivalent when r = . Assume that (.) holdsfor r = . Since∑∞i= iP{ai < |X| ≤ ai+} =∑∞n= P{|X| > an} < ∞, for any ﬁxed ε > , thereexists a positive integer N such that∑∞i=N+ iP{ai < |X| ≤ ai+} < ε. Then, for n >N + ,∣∣∣∣∣a–n · nEXI(|X| ≤ an)– a–nn∑k=EXI(|X| ≤ ak)∣∣∣∣∣≤ a–nn–∑k=E|X|I(ak < |X| ≤ an)= a–nn–∑k=n–∑i=kE|X|I(ai < |X| ≤ ai+)= a–nn–∑i=iE|X|I(ai < |X| ≤ ai+)≤ a–nN∑i=iE|X|I(ai < |X| < ai+)+n–∑i=N+iP{ai < |X| ≤ ai+}Chen et al. Journal of Inequalities and Applications  (2015) 2015:414 Page 8 of 9≤ a–nN∑i=iE|X|I(ai < |X| < ai+)+ ε→ ε as n→ ∞.It follows that∣∣∣∣∣a–n · nEXI(|X| ≤ an)– a–nn∑k=EXI(|X| ≤ ak)∣∣∣∣∣→ as n→ ∞. Hence, to prove (.), by Lemma . it suﬃces to prove thata–nn∑k=(XkI(|X| ≤ ak)– EXkI(|X| ≤ ak)) →  a.s. (.)Since  < an/n/p ↑ and  < p < ,∞∑n=a–n Var(XnI(|X| ≤ an))≤∞∑n=a–n EXI(|X| ≤ an)=∞∑n=a–nn∑k=EXI(ak– < |X| ≤ ak)(set a = )≤∞∑k=akP{ak– < |X| ≤ ak} ∞∑n=ka–n≤ C∞∑k=kP{ak– < |X| ≤ ak}<∞.Then by the Kolmogorov convergence criterion and the Kronecker lemma, (.) holds,and so (.) also holds.Conversely, assume that (.) holds. Let {X ′,X ′n,n ≥ } be an independent copy of{X,Xn,n≥ }. Then we also havea–n( n∑k=X ′k – nbn)→  a.s.Hence, we havea–nn∑k=(Xk –X ′k) →  a.s.So, we have by  < an ↑ thata–n(Xn –X ′n)= a–nn∑k=(Xk –X ′k)–(an–a–n)a–n–n–∑k=(Xk –X ′k) →  a.s.Chen et al. Journal of Inequalities and Applications  (2015) 2015:414 Page 9 of 9By the Borel-Cantelli lemma,∞∑n=P{∣∣Xn –X ′n∣∣ > εan}<∞ ∀ε > ,which, together with (.), implies that (.) holds for r = . So we complete the proof. Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll authors read and approved the manuscript.Author details1Department of Mathematics, Jinan University, Guangzhou, 510630, P.R. China. 2Department of Statistics, Jinan University,Guangzhou, 510630, P.R. China. 3Department of Applied Mathematics, Pai Chai University, Taejon, 302-735, South Korea.AcknowledgementsThe authors would like to thank the referees for the helpful comments. The research of Pingyan Chen and Jiaming Yi issupported by the National Natural Science Foundation of China (No. 11271161). The research of Soo Hak Sung issupported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by theMinistry of Education (2014R1A1A2058041).Received: 1 September 2015 Accepted: 9 December 2015References1. Hsu, PL, Robbins, H: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33, 25-31 (1947)2. Katz, M: The probability in the tail of a distribution. Ann. Math. Stat. 34, 312-318 (1963)3. Baum, LE, Katz, M: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120, 108-123 (1965)4. Lanzinger, H: A Baum-Katz theorem for random variables under exponential moment conditions. Stat. Probab. Lett.39, 89-95 (1998)5. Gut, A, Stadtmüller, U: An intermediate Baum-Katz theorem. Stat. Probab. Lett. 81, 1486-1492 (2011)6. Chen, P, Sung, SH: A Baum-Katz theorem for i.i.d. random variables with higher order moments. Stat. Probab. Lett. 94,63-68 (2014)7. Spa˘taru, A: Generalizing a theorem of Katz. Stat. Probab. Lett. 80, 1136-1140 (2010)8. Li, W, Chen, P, Hu, TC: Complete convergence for moving average processes associated to heavy-tailed distributionsand applications. J. Math. Anal. Appl. 420, 66-76 (2014)9. Sung, SH: On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions.Stat. Probab. Lett. 83, 1963-1968 (2013)10. Petrov, VV: Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press,Oxford (1995)11. Ledoux, M, Talagrand, M: Probability in Banach Spaces. Springer, Berlin (1991)

An extension of the Baum-Katz theorem to i.i.d. random variables with general moment conditions

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An extension of the Baum-Katz theorem to i.i.d. random variables with general moment conditions

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