Equivalent Conditions of Complete Convergence and Complete Moment Convergence for END Random Variables

Transcription

1 Chi. A. Math. Ser. B 391, 2018, DOI: /s Chiese Aals of Mathematics, Series B c The Editorial Office of CAM ad Spriger-Verlag Berli Heidelberg 2018 Equivalet Coditios of Complete Covergece ad Complete Momet Covergece for END Radom Variables Aitig SHEN 1 Mei YAO 2 Beqiog XIAO 3 Abstract I this paper, the complete covergece ad the complete momet covergece for exteded egatively depedet END, i short radom variables without idetical distributio are ivestigated. Uder some suitable coditios, the equivalece betwee the momet of radom variables ad the complete covergece is established. I additio, the equivalece betwee the momet of radom variables ad the complete momet covergece is also proved. As applicatios, the Marcikiewicz-Zygmud-type strog law of large umbers ad the Baum-Katz-type result for END radom variables are established. The results obtaied i this paper exted the correspodig oes for idepedet radom variables ad some depedet radom variables. Keywords Exteded egatively depedet radom variables, Complete covergece, Complete momet covergece 2000 MR Subject Classificatio 60F15 1 Itroductio It is well kow that complete covergece plays a very importat role i the probability limit theory ad mathematical statistics, especially i establishig the strog covergece rate for partial sums of radom variables. The cocept of complete covergece was itroduced by Hsu ad Robbis [1] as follows. Defiitio 1.1 A sequece {U, 1} of radom variables is said to coverge completely to a costat a if for ay ε > 0, P U a > ε <. I this case, we write U a completely. I view of the Borel-Catelli lemma, this implies that U a almost surely a.s., i short. The coverseis true if radom variables{u, 1} Mauscript received November 1, Revised September 6, Correspodig author. School of Mathematical Scieces, Ahui Uiversity, Hefei , Chia. empress201010@126.com 2 School of Mathematics, Hefei Uiversity of Techology, Hefei , Chia; School of Mathematics, Shadog Uiversity, Jia , Chia. ymwalz@163.com 3 School of Mathematical Scieces, Ahui Uiversity, Hefei , Chia @qq.com This work was supported by the Natioal Natural Sciece Foudatio of Chia Nos , , , the Natural Sciece Foudatio of Ahui Provice No J06, the Key Projects for Academic Talet of Ahui Provice No. gxbjzd , the Provicial Natural Sciece Research Project of Ahui Colleges No. KJ2015A018, the Ope Project of School of Mathematical Scieces, Ahui Uiversity No. ADSY201503, the Quality Egieerig Project of Ahui Provice No. 2015jyxm045 ad the Quality Improvemet Projects for Udergraduate Educatio of Ahui Uiversity No. ZLTS

2 84 A. T. She, M. Yao ad B. Q. Xiao are idepedet. Hsu ad Robbis [1] proved that the arithmetic meas of idepedet ad idetically distributed i.i.d., i short radom variables coverges completely to the expected value if the variace of the summads is fiite. Erdös [2] proved the coverse. The result of Hsu-Robbis-Erdös is a fudametal theorem i probability theory ad has bee exteded i several directios by may authors. Oe of the most importat geeralizatios was provided by Baum ad Katz [3] for the strog law of large umbers as follows. Theorem 1.1 Let 1 2 < 1 ad p > 1. Let {X, 1} be idepedet ad idetically distributed radom variables with zero meas. The the followig statemets are equivalet: i E X 1 p < ; ii for all ε > 0, p 2 P X i > ε <. 1.1 Up to ow, there have bee may versios of the Baum-Katz-type results for idepedet ad depedet radom variables, such as Gut [4], Peligrad ad Gut [5], Wag et al. [6], She ad Wu [7], ad so o. Chow [8] geeralized the cocept of complete covergece ad itroduced the cocept of complete momet covergece, which is more geeral tha complete covergece. Let {Z, 1} be a sequece of radom variables, ad a > 0, b > 0, q > 0. If for all ε > 0, the the above result is called the complete momet covergece. a E{b 1 Z ε} q < Chow[8] obtaied the followig result o the complete momet covergece for i.i.d. radom variables. Theorem 1.2 Suppose that {X, 1} is a sequece of i.i.d. radom variables with EX 1 = 0, > 1 2, p 1 ad p > 1. If E[ X 1 p X 1 log1 X 1 ] <, the for all ε > 0, { p 2 E } ε X i <. 1.2 Sice Chow [8] established the result of Theorem 1.2 for i.i.d. radom variables, may authors have studied this type of complete momet covergece for depedet radom variables. See, for example, Che ad Wag [9] for the ϕ-mixig sequece, Wu et al. [10] ad Wag et al. [11] for ρ-mixig sequece ad the martigale differece sequece, respectively, ad so o. We should poit out that the key techiques used i the proofs of Theorems are the Rosethal-type imal momet iequality ad the trucatio methods. All the literatures above adopted these approaches or added extra coditios. There are may sequeces of radom variables satisfyig the Rosethal-type imal momet iequality, such as idepedet radom variables, egatively associated radom variables, egatively supperadditive depedet radom variables, ϕ-mixig radom variables, ρ-mixig radom variables, asymptotically almost egatively associated radom variables, ad so o. But egatively orthat depedet radom variables ad exteded egatively depedet radom variables do ot satisfy the Rosethal-type imal momet iequality. If we wat to geeralize the results of Theorems for i.i.d. radom variables to the case of the exteded egatively depedet settig, we should use differet methods. The mai purpose of this paper is to geeralize the results of Theorems for i.i.d. radom variables to the case of the exteded egatively depedet settig without idetical distributio. I additio, we will preset the sufficiet

3 Equivalet Coditios of Complete Covergece ad Complete Momet Covergece for END 85 ad ecessary coditios of complete momet covergece for exteded egatively depedet radom variables. Now, let us recall the defiitio of exteded egatively depedet radom variables. Defiitio 1.2 A fiite collectio of radom variables X 1,X 2,,X is said to be exteded egatively depedet END, i short, if there exists a costat M > 0 such that both ad PX 1 > x 1,X 2 > x 2,,X > x M PX 1 x 1,X 2 x 2,,X x M PX i > x i PX i x i hold for all real umbers x 1,x 2,,x. A ifiite sequece {X, 1} is said to be END if every fiite subcollectio is END. A array of radom variables {X i,1 i, 1} is called rowwise END radom variables if for every 1, {X i,1 i } are END radom variables. The cocept of the END sequece was itroduced by Liu [12]. I the case M = 1, the otio of END radom variables reduces to the well-kow otio of the so-called egatively orthat depedet NOD, i short radom variables, which was itroduced by Joag-Dev ad Proscha [13]. They also poited out that the egatively associated NA, i short radom variables are NOD ad thus NA radom variables are END. Hece, the class of END icludes the idepedet sequece, the NA sequece ad the NOD sequece as special cases. Studyig the limitig behavior of END radom variables is of great iterest. Some applicatios for the END sequece have bee foud. See, for example, Che et al. [14] established the strog law of large umbers for exted egatively depedet radom variables ad showed its applicatios to risk theory ad reewal theory; She [15] preseted some probability iequalities for END sequeces ad gave some applicatios; Wu ad Gua [16] preseted some covergece properties for the partial sums of END radom variables; Wag ad Wag [17] ivestigated a more geeral precise large deviatio result for radom sums of END real-valued radom variables i the presece of cosistet variatio; Qiu et al. [18] ad Wag et al. [19 21] provided some results o complete covergece for sequeces of END radom variables or arrays of rowwise END radom variables; Wag et al. [22] studied the complete cosistecy for the estimator of oparametric regressio models based o END errors, ad so o forth. The mai purpose of the paper is to geeralize the results of Theorems for i.i.d. radom variables to the case of the END settig, ad the sufficiet ad ecessary coditios of complete momet covergece for END radom variables will also be established. This work is orgaized as follows: Some importat lemmas are provided i Sectio 2. The mai results ad their proofs are preseted i Sectio 3. Throughout this paper, the symbol C deotes a positive costat which is ot ecessarily the same i each appearace, ad a = Ob stads for a = Cb. IA is the idicator fuctio of a evet A. Deote logx = lx,e. 2 Prelimiaries I this sectio, we will provide some importat lemmas, which will be applied to prove the mai results of this paper. The first oe is a basic property for END radom variables, which was give by Liu [23].

4 86 A. T. She, M. Yao ad B. Q. Xiao Lemma 2.1 Let radom variables X 1,X 2,,X be END. If f 1,f 2,,f are all odecreasig or oicreasig fuctios, the radom variables f 1 X 1,f 2 X 2,,f X are END. The ext oe is the Marcikiewicz-Zygmud-type iequality ad the Rosethal-type iequality for partial sums ad imum partial sums of END radom variables. Lemma 2.2 Let p 1 ad {X, 1} be a sequece of END radom variables with EX i = 0 ad E X i p < for each i 1. The there exists a positive costat C p depedig oly o p such that ad E p E X i p E X i p, 1 p 2, 2.1 E p X i p log p E X i p, 1 p 2, 2.2 p [ E X i p E X i p E X i 2 p ] 2, p p [ X i p log p E X i p E X i 2 p 2 ]. 2.4 Proof The iequality 2.3 has bee established by She [15]. The iequality 2.1 ca be obtaied i a similar way as that of Corollary 2.2 i Asadia et al. [24]. The iequalities 2.2 ad 2.4 ca be proved by usig the iequalities 2.1, 2.3 i a similar way as that of Theorem i Stout [25], respectively. The details of the proof are omitted. With Lemma 2.2 accouted for, we ca get the followig importat property for END radom variables, which will play a importat role i provig the mai results of this paper. The proof is similar to that of Lemma A6 i Zhag ad We [26], so the details are omitted. Lemma 2.3 Let {X, 1} be a sequece of END radom variables. The there exists a positive costat C such that for ay x 0 ad all 1, [ ] 2 1 P X k > x P X k > x P X k > x k 1 k k=1 The followig is a basic property for stochastic domiatio. For the proof, oe ca refer to Wu [27], or Wag et al. [6]. Lemma 2.4 Let {X, 1} be a sequece of radom variables, which is stochastically domiated by a radom variable X, i.e., there exists a positive costat C such that P X > x P X > x for all x 0 ad 1. The for ay > 0 ad b > 0, the followig two statemets hold: i E X I X b 1 [E X I X bb P X > b], ii E X I X > b 2 E X I X > b, where C 1 ad C 2 are positive costats. Cosequetly, E X E X, where C is a positive costat.

5 Equivalet Coditios of Complete Covergece ad Complete Momet Covergece for END 87 The last oe comes from Sug [28]. Lemma 2.5 Let Y,Z, 1 be radom variables. The for ay q > 1, ε > 0 ad a > 0, E 1 Y i Z i εa ε q 1 q 1 3 Mai Results ad Their Proofs 1 a q 1E Y i qe Z i. I this sectio, we will give the mai results of this paper, icludig the sufficiet ad ecessary coditios of complete covergece ad complete momet covergece for END radom variables. 3.1 Sufficiet ad ecessary coditios for complete covergece Theorem 3.1 Let > 1 2 ad p > 1. Let {X, 1} be a sequece of END radom variables with EX = 0 if p 1. If there exists a radom variable X ad two positive costats C 1 ad C 2 such that C 1 P X > x P X > x 2 P X > x 3.1 for all x 0 ad 1, the the followig statemets are equivalet: i E X p < ; ii for all ε > 0, p 2 P X i > ε <. 3.2 Proof ii i is trivial. So it suffices to show i ii. We cosider the followig two cases. Case 1 0 < p < 1. For fixed 1, deote, for 1 i, that Y i = X i I X i, Z i = X i I X i >. Notig that X i = Y i Z i, we have that for all ε > 0, p 2 P p 2 P. = H 1 H 2. X i > ε ε Y i > 2 p 2 P ε Z i > It follows from Markov s iequality, C r iequality ad Lemma 2.4, that H 1 p 2 E Y i p 1 [E X I X P X > ]

6 88 A. T. She, M. Yao ad B. Q. Xiao p 1 E X I X CE X p p 1 E X Ik 1 < X k k=1 p 1 k Pk 1 < X k k=1 k p Pk 1 < X k k=1 E X p < 3.4 ad H 2 = C p 2 p 2 E Z i p 2 p 2 1 E X p 2 I X > p 2 1 k= p 2 1 k= E X p 2 Ik < X k 1 k p 2 Pk < X k 1 k p Pk < X k 1 k=1 E X p <. 3.5 Hece, the desired result 3.2 follows from immediately. Case 2 p 1. Notig that p > 1, we take a suitable q such that 1 p < q < 1. For fixed 1, deote for 1 i that Notig that X 1 i = q IX i < q X i I X i q q IX i > q, X 2 i = X i q IX i > q, X 3 i = X i q IX i < q. X i = for 1 j, we have that for all ε > 0, p 2 P X 1 i p 2 P X 2 i X 3 i X i > ε X 1 i > ε 3

7 Equivalet Coditios of Complete Covergece ad Complete Momet Covergece for END 89 p 2 P X 2 i > ε 3 p 2 P X 3 i > ε 3. = I 1 I 2 I Hece, i order to prove 3.2, it suffices to show that I 1 <, I 2 < ad I 3 <. For I 1, we firstly show that EX 1 i It follows from EX = 0, Markov s iequality ad Lemma 2.4, that EX 1 i 0, as. 3.7 [E X i I X i > q q P X i > q ] [E X I X > q q P X > q ] 1q pq E X p I X > q C 1q pq E X p 1q pq E X p, which together with E X p < ad 1 p < q < 1 yields 3.7. Hece, by 3.7, we have that I 1 p 2 P X 1 i EX 1 i ε > For fixed 1, we ca see that {X 1 i,1 i } are still END radom variables by Lemma 2.1. It follows from 3.8, Markov s iequality ad Lemma 2.2, that for ay δ 2, I 1 p 2 δ E i EX 1 [ p 2 δ log δ E X 1 i δ X 1 i EX 1 δ i E X 1 i 2 δ 2 ]. = CI 11 CI Takig δ > { p 1,2,p }, we have 1 2 p 1 δ qδ pq = p δ1 q 1 < 1 ad p 2 δ δ 2 < 1. It follows from C r iequality, Markov s iequality ad Lemma 2.4, that I 11 p 2 δ log δ [E X i δ I X i q qδ P X i > q ]

8 90 A. T. She, M. Yao ad B. Q. Xiao ad < I 12 p 1 δ log δ [E X δ I X q qδ P X > q ] p 1 δqδ pq E X p log δ { p 2 δ log δ [EXi 2 I X i q 2q P X i > q ] p 2 δδ 2 log δ [EX 2 I X q 2q P X > q ] δ 2 C p 2 δδ 2 EX 2 δ 2 log δ, if p 2, C p 2 δδ 2 2 pδ 2 E X p δ 2 log δ, if 1 p < 2 C p 2 δδ 2 EX 2 δ 2 log δ, if p 2, = C p 11 δ 2 1 E X p δ 2 log δ, if 1 p < 2 <. } δ Hece, I 1 < follows from immediately. I the followig, we will show that I 2 <. For fixed 1, deote, for 1 i, that X 4 i = X i q I q < X i q IX i > q. It is easily checked that which implies that X 2 i > ε 3 X i > 1 i I 2 p 2 P X i > p 2 P. = I 21 I 22. X 4 i X 4 i > ε 3 > ε 3, 3.12 It follows from 3.1 ad E X p <, that I 21 p 1 P X > E X p < Notig that 1 p < q < 1, we have from the defiitio of X4 i ad Lemma 2.4, that EX 4 i 1 E X I X > q

9 Equivalet Coditios of Complete Covergece ad Complete Momet Covergece for END 91 Sice X 4 i > 0, by , we have that I 2 1 q pq E X p 0, as p 2 P X 4 i EX 4 i > ε For fixed 1, we ca see that {X 4 i,1 i } are still END radom variables by Lemma 2.1. It follows from Markov s iequality, C r iequality ad Lemma 2.2, that i EX 4 I 2 p 2 δ E X 4 i EX 4 δ i p 2 δ[ E X 4 i δ EX 4 i 2δ 2 ]. = J 1 J By C r iequality ad Lemma 2.4, we ca get that J 1 = C p 2 δ [E X i q δ I q < X i q δ PX i > q ] p 2 δ [E X i δ I X i 2 δ PX i > ] p 2 δ [E X δ I X 2 δ P X > ] p 1 δ E X δ I X 2 CE X p p 1 δ E X δ I2i 1 < X 2i CE X p i δ P2i 1 < X 2i p 1 δ CE X p =i i p P2i 1 < X 2i CE X p E X p < Similarly to the proof of 3.11 ad 3.17, we ca obtai that J 2 <, which, together with 3.16 ad 3.17, yields that I 2 <. Similarly to the proof of I 2 <, oe ca get that I 3 <. Hece, 3.2 follows from 3.6, I 1 <, I 2 < ad I 3 < immediately. This completes the proof of the theorem. With Theorem 3.1 accouted for, we ca get the Marcikiewicz-Zygmud-type strog law of large umbers for END radom variables without idetical distributio as follows. Corollary 3.1 Let > 1 2 ad p > 1. Let {X, 1} be a sequece of END radom variables with EX = 0 if p 1. Assume that there exists a radom variable X ad two

10 92 A. T. She, M. Yao ad B. Q. Xiao positive costats C 1 ad C 2 such that 3.1 holds for all x 0 ad 1. If E X p <, the 1 X i 0 a.s Proof Sice E X p <, by Theorem 3.1, we have that for all ε > 0, p 2 P It follows from 3.19 that, for all ε > 0, > p 2 P X i > ε < X i > ε 2 k1 1 = p 2 P X i > ε k=0 =2 k 2 k p 2 2 k P X i > ε2 k1, if p 2, 1 j 2 k=0 k 2 k1 p 2 2 k P X i > ε2 k1, if 1 p < 2 1 j 2 k=0 k P X i > ε2 k1, if p 2, 1 j 2 k=0 k 1 P X i > ε2 k1, if 1 p < 2, 2 1 j 2 k k=0 which, together with the Borel-Catelli lemma, yields that 1 j 2 k X i 2 k1 0 a.s For all positive itegers, there exists a positive iteger k such that 2 k 1 2 k. We have, by 3.20, that 2 2 X i 1 j 2 k X i X i 0 a.s. 2 k 1 2 k 2 k1 which implies This completes the proof of the corollary. If {X, 1} is a sequece of END radom variables with idetical distributio, the 3.1 is obvious. By usig Theorem 3.1, we ca get the Baum-Katz-type result for END radom variables as follows. Corollary 3.2 Let > 1 2 ad p > 1. Let {X, 1} be a sequece of END radom variables with idetical distributio. Assume further that EX 1 = 0 if p 1. The i ad ii i Theorem 3.1 are equivalet.

11 Equivalet Coditios of Complete Covergece ad Complete Momet Covergece for END Sufficiet ad ecessary coditios for complete momet covergece I this subsectio, we will establish the sufficiet ad ecessary coditios for complete momet covergece. First we preset the ecessary coditio for complete momet covergece. Theorem 3.2 Suppose that the coditios of Theorem 3.1 hold. If for all ε > 0, the E X p <. Proof Note that = ε p 2 E p 2 E p 2 0 ε p 2 0 ε X i <, 3.21 ε X i P p 2 P P ε X i > t dt ε X i > t dt X i > 2ε Combiig 3.21 ad 3.22, we ca get that 3.2 holds for all ε > 0. Hece, E X p < follows from Theorem 3.1 immediately. This completes the proof of the theorem. Next we preset the sufficiet coditio for complete momet covergece. Notig that the factor log is added to the right of the imal momet iequality for END radom variables see Lemma 2.2, i order to establish the sufficiet coditio for complete momet covergece, the momet coditio should be chaged. Theorem 3.3 Suppose that the coditios of Theorem 3.1 hold for p 1. If E X p log θ X < for some θ > { p 1,p}, the for all ε > 0, 1 2 p 2 E ε X i < Proof For fixed 1, deote, for 1 i, that Y i = IX i < X i I X i IX i >, Z i = X i Y i = X i IX i > X i IX i <. It follows from Lemma 2.5, that for ay δ > 1, p 2 E ε X i p 2 δ E Y i EY i δ

12 94 A. T. She, M. Yao ad B. Q. Xiao p 2 E Z i EZ i. = P 1 P Notig that Z i = X i I X i > X i I X i >, we have, by Lemma 2.4, that P 2 = p 2 Z i EZ i = C p 2 E Z i p 2 E X i I X i > p 1 E X I X > E X Ii < X i1 i p 1 C E X Ii < X i1 logi, if p = 1, C E X Ii < X i1 i p, if p > 1 { CE X log X, if p = 1, CE X p, if p > 1 < Next, we will show that P 1 <. Notig that θ > p 1, we ca take δ = θ. We cosider the followig two cases. Case 1 1 < θ 2. It follows from 2.2 of Lemma 2.2 ad Lemma 2.4, that P 1 = C p 2 θ θ E Y i EY i p 2 θ log θ E Y i θ p 2 θ log θ [E X i θ I X i θ P X i > ] p 1 θ log θ [E X θ I X θ P X > ] = C E X θ Ii 1 < X i p 1 θ log θ C Pi < X i1 =i i p 1 log θ

13 Equivalet Coditios of Complete Covergece ad Complete Momet Covergece for END 95 C E X θ Ii 1 < X i i p θ log θ i Pi < X i1 i p log θ i E X p log θ X < Case 2 θ > 2. Sice θ > p 1, we ca see that p 2 θ θ 1 2 < 1. Similarly to the proof of 3.26, we 2 have, by 2.4 of Lemma 2.2, C r iequality ad Lemma 2.4, that P 1 = C p 2 θ E θ Y i EY i [ p 2 θ log θ E Y i θ E Y i 2 θ] 2 C p 2 θ log θ E Y i 2 θ 2 { p 2 θ log θ [EXi 2 I X i 2 P X i > ] p 2 θθ 2 log θ [EX 2 I X 2 P X > ] θ 2 C p 2 θθ 2 EX 2 θ 2 log θ <, if p 2, C p 2 θθ 2 2 pθ 2 E X p θ 2 log θ, if 1 p < 2 C p 2 δθ 2 EX 2 θ 2 log θ, if p 2, = C p 11 θ 2 1 E X p θ 2 log θ, if 1 p < 2 <. }θ Hece, the desired result 3.23 follows from immediately. This completes the proof of the theorem. Ackowledgemets The authors are most grateful to the editor ad aoymous referees for their careful readig of the mauscript ad may valuable suggestios which helped to improve a earlier versio of this paper. Refereces [1] Hsu, P. ad Robbis, H., Complete covergece ad the law of large umbers, Proceedigs of the Natioal Academy of Scieces, 33, 1947, [2] Erdös, O a theorem of Hsu ad Robbis, The Aals of Mathematical Statistics, 20, 1949, [3] Baum, L. E. ad Katz, M., Covergece rates i the law of large umbers, Trasactios of the America Mathematical Society, 1201, 1965,

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