Research Article Moment Inequality for ϕ-mixing Sequences and Its Applications

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1 Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2009, Article ID , 2 pages doi:0.55/2009/ Research Article Momet Iequality for ϕ-mixig Sequeces ad Its Applicatios Wag Xueju, Hu Shuhe, She Ya, ad Yag Wezhi School of Mathematical Sciece, Ahui Uiversity, Hefei, Ahui , Chia Correspodece should be addressed to Hu Shuhe, hushuhe@263.et Received April 2009; Accepted 2 September 2009 Recommeded by Sever Silvestru Dragomir Firstly, the maximal iequality for ϕ-mixig sequeces is give. By usig the maximal iequality, we study the covergece properties for ϕ-mixig sequeces. The Hájek-Réyi-type iequality, strog law of large umbers, strog growth rate ad the itegrability of supremum for ϕ-mixig sequeces are obtaied. Copyright q 2009 Wag Xueju et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.. Itroductio Let {X, } be a radom variable sequece defied o a fixed probability space Ω, F,P ad S i X i for each. Let ad m be positive itegers. Write F m σ X i, i m.giveσ-algebras B, R i F,let ϕ B, R sup P B A P B. A B,B R,P A >0. Defie the ϕ-mixig coefficiets by ϕ sup ϕ k F k, F k, 0..2 Defiitio.. A radom variable sequece {X, } is said to be a ϕ-mixig radom variable sequece if ϕ 0as. The cocept of ϕ-mixig radom variables was itroduced by Dobrushi ad may applicatios have bee foud. See, for example, Dobrushi,Utev 2, ad Che 3

2 2 Joural of Iequalities ad Applicatios for cetral limit theorem, Herrdorf 4 ad Peligrad 5 for weak ivariace priciple, Se 6, 7 for weak covergece of empirical processes, Iosifescu 8 for limit theorem, Peligrad 9 for Ibragimov-Iosifescu cojecture, Shao 0 for almost sure ivariace priciples, Hu ad Wag for large deviatios, ad so forth. Whe these are compared with the correspodig results of idepedet radom variable sequeces, there still remais much to be desired. The mai purpose of this paper is to study the maximal iequality for ϕ-mixig sequeces, by which we ca get the Hájek-Réyi-type iequality, strog law of large umbers, strog growth rate, ad the itegrability of supremum for ϕ-mixig sequeces. Throughout the paper, C deotes a positive costat which may be differet i various places. The mai results of this paper deped o the followig lemmas. Lemma.2 see Lu ad Li 2. Let {X, } be a sequece of ϕ-mixig radom variables. Let X L p F k, Y L q F, p, q, ad /p /q. The k EXY EXEY 2 ϕ /p E X p /p E Y q /q..3 Lemma.3 see Shao 0. Let {X, } be a ϕ-mixig sequece. Put T a a i a X i. Suppose that there exists a array {C a, } of positive umbers such that ET 2 a C a, for every a 0,..4 The for every q 2, there exists a costat C depedig oly o q ad ϕ such that E max Ta j [ ] q C C q/2 a, E max X i q j a i a.5 for every a 0 ad. Lemma.4 see Hu et al. 3. Let b,b 2,... be a odecreasig ubouded sequece of positive umbers ad let α,α 2,... be oegative umbers. Let r ad C be fixed positive umbers. Assume that for each, the r E max S l C l b r l l S, l.6 <,.7 lim 0 a.s.,.8 ad with the growth rate S O β a.s.,.9

3 Joural of Iequalities ad Applicatios 3 where β max k b kv δ/r k, 0 <δ<, v E max S l r 4C l b l b r l l S l r 4C l b l α k b r k k b r l l <, <. β, lim 0,.0 If further we assume that α > 0 for ifiitely may,the S l r 4C l β l β r l l <.. Lemma.5 see Fazekas ad Klesov 4 ad Hu 5. Let b,b 2,... be a odecreasig ubouded sequece of positive umbers ad let α,α 2,... be oegative umbers. Deote Λ k α α 2 α k for k. Letr be a fixed positive umber satisfyig.6.if l Λ l b r l b r <,.2 l Λ b r is bouded,.3 the.8. hold. 2. Maximal Iequality for ϕ-mixig Sequeces Theorem 2.. Let {X, } be a sequece of ϕ-mixig radom variables satisfyig ϕ/2 <. Assume that EX 0 ad EX 2 < for each. The there exists a costat C depedig oly o ϕ such that for ay ad a 0 E a 2 X i C i a E max a j j i a a i a X i 2 C EX 2 i, a i a EX 2 i

4 4 Joural of Iequalities ad Applicatios I particular, 2 E X i C EX 2 i, i i j 2 E max X i C EX 2 i, j i i where C may be differet i various places. Proof. By Lemma.2 for p q 2, we ca see that E a 2 X i i a a i a a i a EX 2 i 2 EX 2 i 4 a i<j a a i<j a E X i ϕ /2 j i EX 2 i /2 /2 EX 2 j a i a EX 2 i 2 4 ϕ /2 k C a i a k EX 2 i, a k k i a a EX 2 i i a ϕ /2 k EX 2 i EX 2 k i 2.5 which implies 2.. By 2. ad Lemma.3 take q 2, we ca get E max a j j i a X i 2 C C [ a i a a i a EX 2 i E EX 2 i. max a i a X2 i ] 2.6 The proof is completed. 3. Hájek-Réyi-Type Iequality for ϕ-mixig Sequeces I this sectio, we will give the Hájek-Réyi-type iequality for ϕ-mixig sequeces, which ca be applied to obtai the strog law of large umbers ad the itegrability of supremum. Theorem 3.. Let {X, } be a sequece of ϕ-mixig radom variables satisfyig ϕ/2 < ad let {, } be a odecreasig sequece of positive umbers. The for

5 Joural of Iequalities ad Applicatios 5 ay ε>0 ad ay iteger, P max Xj E k b k ε 4C Var, 3. ε 2 j j where C is defied i 2.4 i Theorem 2.. Proof. Without loss of geerality, we assume that for all. Let α 2. For i 0, defie A i { k : α i b k <α i }. 3.2 For A i /, weletv i max{k : k A i } ad t be the idex of the last oempty set A i. Obviously, A i A j if i / j ad t i 0 A i {, 2,...,}. It is easily see that α i b k b v i <α i if k A i ad {X EX, } is also a sequece of ϕ-mixig radom variables. By Markov s iequality ad 2.4, we have P max Xj E k b k j ε P max max 0 i t,a i / k A i Xj E b k j ε t P max α i Xj E k v i ε i 0,A i / ε 2 C ε 2 C ε 2 t i 0,A i / t i 0,A i α / 2i j j α E 2i max 2 Xj E k v i Var j v i Var t j i 0,A i /,v i j α. 2i 3.3 Now we estimate t i 0,A i /,v i j /α2i.leti 0 mi{i : A i /, v i j}. The b j b v i0 < α i 0 follows from the defiitio of v i. Therefore, t i 0,A i /,v i j < α 2i i i 0 α 2i /α 2 α 2i 0 < α 2 /α 2 4. b j Thus, 3. follows from 3.3 ad 3.4 immediately.

6 6 Joural of Iequalities ad Applicatios Theorem 3.2. Let {X, } be a sequece of ϕ-mixig radom variables satisfyig ϕ/2 < ad let {, } be a odecreasig sequece of positive umbers. The for ay ε>0ad ay positive itegers m<, P max Xj E m k b k ε 4C m Var 4 ε 2 j j b 2 m j m Var, 3.5 where C is defied i 3.. Proof. Observe that max Xj E m k b k j b m j m Xj E max m k b k Xj E, j m 3.6 thus P max Xj E m k b k ε j P m Xj E ε P max 2 m k b k b m j Xj E ε I II. 2 j m 3.7 For I, by Markov s iequality ad 2.3, we have I 4 E m Xj EX ε 2 bm 2 j j 2 4C ε 2 m Var. 3.8 j b 2 m For II, we will apply Theorem 3. to {X m i, i m} ad {b m i, i m}. Notig that max m k b k Xj E j m max k m b m k j Xm j EX m j, 3.9 thus, by Theorem 3., weget II 4C m Var X m j ε/2 2 j b 2 m j 6C ε 2 j m Var. 3.0 Therefore, the desired result 3.5 follows from immediately.

7 Joural of Iequalities ad Applicatios 7 Theorem 3.3. Let {X, } be a sequece of ϕ-mixig radom variables satisfyig ϕ/2 < ad let {, } be a odecreasig sequece of positive umbers. Deote T i X i EX i for. Assume that Var <, 3. j the for ay r 0, 2, T r 4Cr Var <, r j where C is defied i 3.. Furthermore, if lim,the lim j Xj E 0 a.s. 3.3 Proof. By the cotiuity of probability ad Theorem 3., weget T r P sup b T r 0 b >t dt P sup T r 0 b >t dt P sup T r b P sup T b dt >t/r lim P max N T dt >t/r Var 4C j 4Cr 2 r N t 2/r dt Var <. j >t dt 3.4 Observe that T P >ε P max T >ε lim P max N T. >ε 3.5 m N m m N m N

8 8 Joural of Iequalities ad Applicatios By Theorem 3.2, we have that P max m N T >ε 4C ε 2 m Var 4 j b 2 m N j m Var. 3.6 Hece, by 3. ad Kroecker s Lemma, it follows that lim P T m >ε 0, ε >0, 3.7 m which is equivalet to lim j Xj E 0 a.s. 3.8 So the desired results are proved. 4. Strog Law of Large Numbers ad Growth Rate for ϕ-mixig Sequeces Theorem 4.. Let {X, } be a sequece of mea zero ϕ-mixig radom variables satisfyig ϕ/2 < ad let {, } be a odecreasig ubouded sequece of positive umbers. Assume that EX 2 b 2 <, 4. the S lim 0 a.s., 4.2 ad with the growth rate S O β a.s., 4.3

9 Joural of Iequalities ad Applicatios 9 where β max b kv δ/2 α k β, 0 <δ<, v k k, lim 0, k b 2 b k 4.4 α k CEX 2 k, k, where C is defied i 2.4, E max S l 2 4 l b l l b 2 l S l 2 4 l b l l b 2 l <, <. 4.5 If further we assume that α > 0 for ifiitely may,the S l 2 4 l β l l β 2 l <. 4.6 Proof. By 2.4 i Theorem 2., we have E max S k 2 k C EX 2 i i α k. k 4.7 It follows by 4. that α b 2 C EX 2 b 2 <. 4.8 Thus, follow from 4.7, 4.8,adLemma.4 immediately. We complete the proof of the theorem. Theorem 4.2. Let {X, } be a sequece of ϕ-mixig radom variables with ϕ/2 <. p<2. Deote Q max k EX 2 k for ad Q 0 0. Assume that Q <, 2/p 4.9 the lim /p i X i EX i 0 a.s., 4.0

10 0 Joural of Iequalities ad Applicatios ad with the growth rate /p i X i EX i O β /p a.s., 4. where β max k k/p v δ/2 k, 0 <δ<, v k α k k 2/p, lim β 0, /p α k C kq k k Q k, k, where C is defied i 2.4, E max S l 2 4 <, l l /p l l 4.3 2/p S l 2 4 <. l /p l 4.4 2/p l l 4.2 If further we assume that α > 0 for ifiitely may,the S l 2 4 l β l l β 2 l <. 4.5 I additio, for ay r 0, 2, S l r 4r 2 r l l /p l <. l 4.6 2/p Proof. Assume that EX 0, /p, ad Λ l,. By 2.4 i Theorem 2., we ca see that 2 E max X i C EX 2 i CQ k i i α k. k 4.7 It is a simple fact that α k 0 for all k. It follows by 4.9 that l Λ l b 2 l b 2 l C 2C p l lq l l 2/p l 2/p l Q l <. l2/p 4.8

11 Joural of Iequalities ad Applicatios That is to say.2 holds. By Remark 2. i Fazekas ad Klesov 4,.2 implies.3. By Lemma.5, we ca obtai immetiately. By 4.4, it follows that S l r P sup l l /p S l r 0 l l /p >t dt P sup S l dt >t/r The proof is completed. l l l /p l /p S l 2 t 2/r dt 4r 2 r l <. l2/p 4.9 Remark 4.3. By usig the maximal iequality, we get the Hájek-Réyi-type iequality, the strog law of large umbers ad the strog growth rate for ϕ-mixig sequeces. I additio, we get some ew bouds of probability iequalities for ϕ-mixig sequeces, such as 3., 3.5, 3.2, ,ad Ackowledgmets The authors are most grateful to the Editor Sever Silvestru Dragomir, the referee Professor Mihaly Becze ad a aoymous referee for careful readig of the mauscript ad valuable suggestios ad commets which helped to sigificatly improve a earlier versio of this paper. This work was supported by the Natioal Natural Sciece Foudatio of Chia Grat os , ad the Iovatio Group Foudatio of Ahui Uiversity. Refereces R. L. Dobrushi, The cetral limit theorem for o-statioary Markov chai, Theory of Probability ad Its Applicatios, vol., o. 4, pp , S. A. Utev, O the cetral limit theorem for ϕ-mixig arrays of radom variables, Theory of Probability ad Its Applicatios, vol. 35, o., pp. 3 39, D. C. Che, A uiform cetral limit theorem for ouiform ϕ-mixig radom fields, The Aals of Probability, vol. 9, o. 2, pp , N. Herrdorf, The ivariace priciple for ϕ-mixig sequeces, Zeitschrift für Wahrscheilichkeitstheorie ud Verwadte Gebiete, vol. 63, o., pp , M. Peligrad, A ivariace priciple for ϕ-mixig sequeces, The Aals of Probability, vol. 3, o. 4, pp , P. K. Se, A ote o weak covergece of empirical processes for sequeces of ϕ-mixig radom variables, Aals of Mathematical Statistics, vol. 42, o. 6, pp , P. K. Se, Weak covergece of multidimesioal empirical processes for statioary ϕ-mixig processes, The Aals of Probability, vol. 2, o., pp , J. Iosifescu, Limit theorem for ϕ-mixig sequeces, i Proceedigs of the 5th Coferece o Probability Theory, pp. 6, September M. Peligrad, O Ibragimov-Iosifescu cojecture for ϕ-mixig sequeces, Stochastic Processes ad Their Applicatios, vol. 35, o. 2, pp , 990.

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