MAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES
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1 Commu Korea Math Soc 26 20, No, pp 5 6 DOI 0434/CKMS20265 MAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES Wag Xueju, Hu Shuhe, Li Xiaoqi, ad Yag Wezhi Abstract Let {X, } be a sequece of asymptotically almost egatively associated radom variables ad S = X i I the paper, we get the precise results of Hájek-Réyi type iequalities for the partial sums of asymptotically almost egatively associated sequece, which geeralize ad improve the results of Theorem 24 Theorem 26 i Ko et al [4] I additio, the large deviatio of S for sequece of asymptotically almost egatively associated radom variables is studied At last, the Marcikiewicz type strog law of large umbers is give Itroductio Defiitio A fiite collectio of radom variables X, X 2,, X is said to be egatively associated NA, i short if for every pair of disjoit subsets A, A 2 of {, 2,, }, Cov{fX i : i A, g : j A 2 } 0, wheever f ad g are coordiatewise odecreasig such that this covariace exists A ifiite sequece {X, } is NA if every fiite subcollectio is NA Defiitio 2 A sequece {X, } of radom variables is called asymptotically almost egatively associated AANA, i short if there exists a oegative sequece q 0 as such that CovfX, gx +, X +2,, X +k q [V arfx V argx +, X +2,, X +k ] /2 Received September 27, Mathematics Subject Classificatio 60E0, 60F5, 60F0 Key words ad phrases Hájek-Réyi iequality, asymptotically almost egatively associated sequece, strog law of large umbers, large deviatio Supported by the NNSF of Chia 08700, , Provicial Natural Sciece Research Project of Ahui Colleges KJ200A005, Talets Youth Fud of Ahui Provice Uiversities 200SQRL06ZD, Youth Sciece Research Fud of Ahui Uiversity 2009QN0A 5 c 20 The Korea Mathematical Society
2 52 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI for all, k ad for all coordiatewise odecreasig cotiuous fuctios f ad g wheever the variaces exist The family of AANA sequece cotais NA ad idepedet sequeces as special cases A AANA sequece of radom variables meas roughly that asymptotically the future is almost egatively associated with the preset A example of a AANA sequece which is ot NA was costructed by Chadra ad Ghosal [] Sice the cocept of AANA sequece was itroduced by Chadra ad Ghosal [], may applicatios have bee foud See for example, Chadra ad Ghosal [] derived the Kolmogorov type iequality ad the strog law of large umbers, Chadra ad Ghosal [2] obtaied the almost sure covergece of weighted averages, Ko et al [4] studied the Hájek-Réyi type iequality, ad Wag et al [5] established the law of the iterated logarithm for product sums Recetly, Yua ad A [6] established some Rosethal type iequalities for imum partial sums of AANA sequeces The mai purpose of the paper is to further study the Hájek-Réyi type iequalities, which geeralize ad improve the results of Theorem 24 Theorem 26 i Ko et al [4] I additio, the large deviatio ad Marcikiewicz type strog law of large umbers for AANA sequece are studied Throughout the paper, let {X, } be a sequece of AANA radom variables defied o a fixed probability space Ω, F, P Deote S = X i ad IA be the idicator fuctio of the set A For p >, let q = p/p be the dual umber of p C deotes a positive costat which may be differet i various places Lemma cf Yua ad A, [6, Lemma 2] Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, }, f, f 2, be all odecreasig or oicreasig fuctios, the {f X, } is still a sequece of AANA radom variables with mixig coefficiets {q, } Lemma 2 Let < p 2 ad {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad EX = 0 for each If = q2 <, the there exists a positive costat C p depedig oly o p such that 2 E S i p C p E X i p i for all, where C p = 2 p [ 2 2 p p + 6p p = q2 p/q ] We poit out that Lemma 2 has bee studied by Yua ad A [6] But here we give the accurate coefficiet C p Ad Lemma 2 geeralizes ad improves the result of Lemma 22 i Ko et al [4] The followig Khitchie- Kolmogorov type covergece theorem is the immediate byproduct of Lemma ad Lemma 2
3 MAXIMAL INEQUALITIES AND STRONG LAW 53 Corollary Khitchie-Kolmogorov type covergece theorem Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad = q2 < Assume that 3 V arx <, = the = X EX coverges as Lemma 3 cf Fazekas ad Klesov, [3, Theorem ] Let β, β 2,, β be a odecreasig sequece of positive umbers ad α, α 2,, α be oegative umbers Let r be a fixed positive umber Assume that for each m with m, r l m 4 E l m α l, β l l= the l 5 E X r j 4 l α l β r l= l Lemma 4 cf Yua ad A, [6, Theorem 2] Let {X, } be a sequece of AANA radom variables with EX i = 0 for all i ad p 3 2 k, 4 2 k ], where iteger umber k If = qq/p <, the there exists a positive costat D p depedig oly o p such that for all p/2 6 E S i p D p E X i i p + EXi 2 2 Hájek-Réyi type iequalities for AANA sequece Theorem 2 Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad {b, } be a odecreasig sequece of positive umbers Assume that EX = 0 for each ad = q2 < The for ay ε > 0 ad ay iteger, 2 P k ε 2p C p E p ε p b p j [ for all < p 2, where C p = 2 p 2 2 p p + 6p p = q2 ] p/q Proof Without loss of geerality, we assume that b 0 = 0 ad i = 0 whe i = It is easy to check that 22 S k = = j b i b i = b i b i j=i
4 54 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI 22 ad k b k b i b i = imply that S k 23 b k ε i k Therefore k S k b k which implies that 24 P k ε S k b k = j=i k i k j=i i k i ε ε i b j b j ε i ε, 2 ε i P i ε 2 By Lemma, we ca see that {X /b, } is still a sequece of AANA radom variables with mixig coefficiets {q, } Thus, by 24, Markov s iequality ad Lemma 2, we ca obtai P k ε 2p ε p E i i The proof of the theorem is complete p 2p C p ε p E p Theorem 22 Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad {b, } be a odecreasig sequece of positive umbers Assume that EX = 0 for each ad = q2 < The for ay ε > 0 ad ay positive itegers m <, 25 P m k ε 2p C p m E p ε p b p + 2 p E p m b p j=m+ j [ for all < p 2, where C p = 2 p 2 2 p p + 6p p = q2 ] p/q Proof Observe that m k m b m + m+ k j=m+, b p j
5 MAXIMAL INEQUALITIES AND STRONG LAW 55 thus 26 P m k ε P m b m ε + P 2 = I + II m+ k j=m+ For I, by Markov s iequality ad Lemma 2, we have p 27 I 2p m ε p b p E 2p C p m E p m ε p b p m ε 2 For II, we will apply Theorem 2 to {X m+i, i m} ad {b m+i, i m} Notig that m+ k = k m X m+j b m+k, j=m+ thus, by Theorem 2, we get 28 II 2p m C p E X m+j p ε/2 p b p = 22p C p m+j ε p j=m+ E p Therefore, the desired result 25 follows from immediately b p j Theorem 23 Let {X, } be a sequece of AANA radom variables with mixig coefficiets {q, } ad = q2 <, {b, } be a odecreasig sequece of positive umbers Deote T = X i EX i for Assume that V ar 29 b 2 < j The for ay r 0, 2, 20 E sup T b ad 2 E sup r T b + 4rC 2 2 r 2 4C 2 V ar b 2 j V ar b 2 j < <
6 56 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI Furthermore, if lim b = +, the lim b E = 0 as, where 2/2 C 2 = q 2 = q 2 < = Proof By the cotiuity of probability ad Theorem 2, we ca see that E sup T r r b = P sup T r 0 b > t dt + P sup T b > t dt + lim P T N > t/r dt + 4C 2 = + 4rC 2 2 r V ar b 2 j N b V ar b 2 j t 2/r dt < So 20 is proved By Lemma 2, we have 22 E T i 2 C 2 E X i EX i 2 = C 2 i = V arx i = where α j = C 2 V ar 0, j =, 2,, By 22 ad Lemma 3, 23 E T i 2 α j V ar i 4 b i b 2 = 4C 2 j b 2 j Thus, by mootoe covergece theorem ad 23, E sup T 2 [ = E T i 2] = lim b 4C 2 lim i V ar b 2 j b i < This completes the proof of 2 Observe that T P > ε = P =m b N=m = lim N P m N m N By Theorem 22 for p = 2 we ca obtai that P T m N b > ε 4C m 2 V ar ε 2 b m E T b T b N j=m+ α j, T i 2 i b i > ε > ε V ar b 2 j
7 MAXIMAL INEQUALITIES AND STRONG LAW 57 Hece, by 29 ad Kroecker s lemma, it follows that lim P T m > ε = 0, ε > 0, which is equivalet to lim are proved =m b b E = 0 as The desired results Remark 2 Hájek-Réyi type iequalities for AANA sequece have bee studied by Ko et al [4] But their results are based o the followig coditios 24 k= σ M/M k /M D k= σ 2 k /2 for some M >, D > 0, ad EXk 2 <, where σ2 k = EX2 k Here Theorem 2 Theorem 23 we remove the coditios above, ad geeralize p = 2 to the case of < p 2 I additio, we give the accurate coefficiet C p So our Theorem 2 Theorem 23 geeralize ad improve the results of Theorem 24 Theorem 26 i Ko et al [4], respectively 3 Large deviatios for AANA sequece I this sectio, we will study the asymptotic behavior of the probabilities 3 P S > x, x > 0, I the followig, we let X p = E X p /p for some p > 0 Theorem 3 Let < p 2 ad {X, } be a sequece of AANA radom variables with = q2 < ad EX i = 0 for all i If there exists a positive costat M < such that X i p M for all i, the for every x > 0, 32 P S i > x C pm p i x p p, where C p is defied i Lemma 2 Proof By Markov s iequality ad Lemma 2, we ca see that P S i > x i p x p E S i p i 33 which implies 32 C p p x p E X i p C pm p x p p,
8 58 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI Theorem 32 Let {X, } be a sequece of AANA radom variables with EX i = 0 for all i If there exists a positive costat M < such that X i p M for all i ad some p 3 2 k, 4 2 k ], where iteger umber k, the for every x > 0, P S i > x 2D pm p i x p p/2, where D p is defied i Lemma 4 Proof By 0 < 2/p < ad C r s iequality, 2/p X i p which implies that Xi 2, p/2 E X i p E Xi 2 By Jese s iequality, we have p/2 EXi 2 E p/2 Xi 2 Therefore, the statemets above ad C r s iequality imply that p/2 35 E X i p + 2 p/2 E X i p 2M p p/2 EX 2 i Combiig Lemma 4 ad 35, 36 E S i p 2D p M p p/2 i It follows from Markov s iequality ad 36 that 37 P S i > x i p x p E this completes the proof of the theorem S i p i 2D pm p x p p/2, 4 Marcikiewicz type strog law of large umbers for AANA sequece Theorem 4 Let {X, } be a sequece of idetically distributed AANA radom variables with = q2 < ad E X p < for 0 < p < 2 Assume that EX = 0 if p < 2 The 4 X /p k 0 as, k=
9 MAXIMAL INEQUALITIES AND STRONG LAW 59 Proof Deote the Y = /p IX /p + X I X < /p + /p IX /p, P X Y = P X /p CE X p <, = = which implies that P X Y, io = 0 by the Borel-Catelli lemma Thus /p k= X k 0 as if ad oly if /p k= Y k 0 as So we oly eed to show that 42 Y /p k EY k 0 as,, ad 43 k= /p EY k 0, k= By Corollary ad Kroecker s lemma, to prove 42, it suffices to show that Y 44 V ar < I fact, V ar = Y /p = = /p C P X /p + C C + C = C + C C + C < = 2/p k= =k = EX 2 I X < /p 2/p EX 2 Ik X p < k k= E X p X 2/p 2 p Ik X p < k k 2/p E X p k 2 p/p Ik X p < k k= Hece 42 holds Next, we will prove 43 It will be divided ito two cases: i If p =, by E X p < ad Lebesgue domiated covergece theorem, we have 45 lim /p P X /p = 0, lim 46 EX I X < /p = lim X ωi X ω < /p P dω Ω = EX = 0
10 60 WANG XUEJUN, HU SHUHE, LI XIAOQIN, AND YANG WENZHI Thus, EY /p P X /p + EX I X < /p 0 as By the Toeplitz lemma, we obtai lim k= EY k = 0 ii If p, by the Kroecker s lemma, to prove 43, it suffices to show that 47 For 0 < p <, = = = EY < /p EY P X /p /p + C + = C + C + C = =j = E X I X < /p /p /p E X Ij X p < j /p E X Ij X p < j j /p E X p j p/p Ij X p < j < For p < 2, by EX = 0, we ca see that EY P X /p /p + = = C + = C + C + C = /p E X I X /p = = EX I X < /p /p j /p E X Ij X p < j + j /p E X p j p/p Ij X p < j + < Thus 47 holds, which implies 43 by Kroecker s lemma We get the desired result Ackowledgemets The authors are most grateful to the editor ad aoymous referees for careful readig of the mauscript ad valuable suggestios which helped i improvig a earlier versio of this paper
11 MAXIMAL INEQUALITIES AND STRONG LAW 6 Refereces [] T K Chadra ad S Ghosal, Extesios of the strog law of large umbers of Marcikiewicz ad Zygmud for depedet variables, Acta Math Hugar 7 996, o 4, [2], The strog law of large umbers for weighted averages uder depedece assumptios, J Theoret Probab 9 996, o 3, [3] I Fazekas ad O Klesov, A geeral approach to the strog laws of large umbers, Teor Veroyatost i Primee , o 3, ; traslatio i Theory Probab Appl , o 3, [4] M H Ko, T S Kim, ad Z Y Li, The Hájeck-Rèyi iequality for the AANA radom variables ad its applicatios, Taiwaese J Math , o, 22 [5] Y B Wag, J G Ya, ad F Y Cheg, The strog law of large umbers ad the law of the iterated logarithm for product sums of NA ad AANA radom variables, Southeast Asia Bull Math , o 2, [6] D M Yua ad J A, Rosethal type iequalities for asymptotically almost egatively associated radom variables ad applicatios, Sci Chia Ser A , o 9, Wag Xueju School of Mathematical Sciece Ahui Uiversity Hefei , P R Chia address: wxjahdx2000@26com Hu Shuhe School of Mathematical Sciece Ahui Uiversity Hefei , P R Chia address: hushuhe@263et Li Xiaoqi School of Mathematical Sciece Ahui Uiversity Hefei , P R Chia address: lixiaoqi983@63com Yag Wezhi School of Mathematical Sciece Ahui Uiversity Hefei , P R Chia address: wzyag827@63com
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