Precise Rates in Complete Moment Convergence for Negatively Associated Sequences

Transcription

1 Commuicatios of the Korea Statistical Society 29, Vol. 16, No. 5, Precise Rates i Complete Momet Covergece for Negatively Associated Sequeces Dae-Hee Ryu 1,a a Departmet of Computer Sciece, ChugWoo Uiversity Abstract Let {X, 1} be a egatively associated sequece of idetically distributed radom variables with mea zeros ad positive fiite variaces. Set S = i=1 X i. Suppose that < σ 2 = EX i=2 CovX 1, X i ) <. We prove that, if EX 2 1 log+ X 1 ) δ < for ay < δ 1, the log ) δ 1 =2 2 ES I 2 S ɛσ log ) = E N 2δ+2, δ where N is the stadard ormal radom variable. We also prove that if S is replaced by M = max 1 k S k, the the precise rate still holds. Some results i Fu ad Zhag 27) are improved to the complete momet case. Keywords: Precise rates, complete momet covergece, egatively associated, law of the logarithm. 1. Itroductio A fiite sequece of radom variables {X i, 1 i } is said to be egatively associatedna), if for every disjoit subsets A ad B of {1, 2,..., }, we have Cov f X i ; i A), gx j ; j B)), wheever f o R A ad g o R B are coordiatewise odecreasig fuctios ad the covariace exists. A ifiite sequece of radom variables is NA if every fiite subsequece is NA. The otio of NA was itroduced by Alam ad Saxea 1981). Joag-Dev ad Proscha 1983) showed that may well kow multivariate distributios possess the NA property. Some examples iclude: a) the multiomial, b) the covolutio of ulike multiomials, c) the multivariate hypergeometric distributio, d) the Dirichlet, e) the Dirichlet compoud multiomial, f) the egatively correlated ormal distributio, g) the permutatio distributio, h) the radom samplig without replacemet, ad i) the joit distributio of raks. Because of its wide applicatios i multivariate statistical aalysis ad system reliability, the otio of NA has received cosiderable attetio recetly. We refer to Joag-Dev ad Proscha 1983) for fudametal properties, Shao ad Su 1999) for the law of the iterated logarithm, Shao 2) for momet iequalities ad the maximal iequalities of the partial sum, Liag 2) for complete covergece, Kim et al. 21) for the estimatio of empirical distributio, Li ad Zhag 24) for complete momet covergece, Fu ad Zhag 27) for the precise rates of i the law of the logarithm ad Ko 29) for cetral limit theorem of a liear process based o the egatively associated process i a Hilbert space. Set S = i=1 X i ad deote log x = lx e). Whe {X, 1} is a sequece of i.i.d. radom variables Liu ad Li 26) proved as follows: Let {X, 1} be a sequece of i.i.d. radom variables. Suppose that EX 1 =, < EX1 2 = σ2 ad EX1 2 log + X 1 )δ <, 1.1) 1 Professor, Departmet of Computer Sciece, ChugWoo Uiversity, Chugam , Korea. rdh@chugwoo.ac.kr

2 842 Dae-Hee Ryu for < δ 1. The log ) δ 1 =2 where N is the stadard ormal radom variable. Coversely, if 1.2) is true, the 1.1) holds. Let where 2 ρ) =: sup k 1 ES I 2 S ɛ log ) = σ2+2δ E N 2δ+2, 1.2) δ sup X L 2 Fk ) Y L 2 F + EXY EXEY sup, k+ ) VarX)VarY) F = σx i ; 1 i ) ad F + = σx i ; i ). The the sequece {X, 1} is said to be ρ-mixig, if ρ) as see e.g. Peligrad, 1987). Zhao 28) proved that 1.2) is true for ρ-mixig sequeces uder appropriate coditios as follows: Let {X, 1} be a strictly statioary sequece of ρ-mixig radom variables with EX 1 = ad EX1 2 <. Suppose that ES 2 lim = σ 2 >, ρ 2 q 2 ) <, 1.3) =1 for q > 2δ + 2 ad EX 2 1 log+ X 1 ) δ < for ay < δ 1. The log ) δ 1 =2 2 ES I 2 S ɛσ log ) = E N 2δ ) δ Furthermore, Zhao 28) proved that if S is replaced by M = max 1 k S k the the precise rate still holds as follows: Let {X, 1} be a strictly statioary sequece of ρ-mixig radom variables satisfyig above coditios. The, log ) δ 1 =2 2 EM 2 I M ɛσ log ) = 2E N 2δ+2 δ = 1). 1.5) 2 + 1) 2δ+2 The purpose of this paper is to show that, for egatively associated radom variables 1.4) ad 1.5) still hold uder appropriate coditios. 2. Prelimiaries We itroduce some prelimiary results which are eeded i provig the mai results. Lemma 1. Newma, 1984) Assume that {X, 1} is a strictly statioary sequece of egatively associated radom variables with EX 1 = ad EX1 2 <. If < σ2 = EX i=2 CovX 1, X i ) <, the S σ D N, 1) as, 2.1) where D idicates covergece i distributio, N a stadard ormal distributio.

3 Precise Rates i Complete Momet Covergece for NA Sequeces 843 Lemma 2. Shao, 2) Let {X, 1} be a strictly statioary sequece of egatively associated radom variables with EX 1 = ad EX1 2 <. If < σ2 = EX i=2 CovX 1, X i ) <, the W W, where W is a stadard Wieer measure, W t) = S [t] /σ, t 1, ad meas weak covergece i D[, 1] with Skorohod topology. I particular, M σ D sup Wt), 2.2) t 1 where M = max 1 k S k, 1 ad {Wt); t } is a stadard Wieer process. Lemma 3. Shao, 2) Let {Y i, 1 i } be a sequece of NA radom variables with mea zeros ad fiite variaces. Deote S k = k i=1 Y i, 1 k, B = i=1 EYi 2. The, for ay u >, v >. ) ) ) { } u P max S k u 2P max Y k v + 4 exp u2 12v B ) 1 k 1 k 8B 4uv + B ) Propositio 1. Zhao, 28) Suppose that N is a stadard ormal radom variable. The for ay < δ 1, +2 log ) δ P N ɛ log ) = E N 2δ+2 δ ) Proof: For the proof see Theorem 1.3 i Huag ad Zag 25). =2 Propositio 2. Zhao, 28) Suppose that N is a stadard ormal radom variable. The for ay < δ 1, log ) δ 1 2xP N x ) = E N 2δ+2 δδ + 1). 2.5) =2 2 ɛ log Proof: See the proof of Propositio 5.1 i Liu ad Li 26). The followig result is similar to oe of Propositio 3.2 i Fu ad Zhag 27). Propositio 3. Let {X ; 1} be a egatively associated sequece of idetically distributed radom variables with EX 1 = ad EX1 2 <. Suppose that < σ2 = EX i=2 CovX 1, X i ) = 1, ad EX1 2log+ X 1 ) δ < for ay < δ 1. The +2 log ) δ P S ɛ log ) P N ɛ log ) =. 2.6) =2 Proof: Usig the stadard method, set Hɛ) = [expm/ɛ 2 )], where M > 4, < ɛ < 1/4. I fact, we get log ) δ P S ɛ log ) P N ɛ log ) =2 log ) δ = P S ɛ log ) P N ɛ log ) + log ) δ P S ɛ log ) P N ɛ log ) =: I + II. 2.7)

4 844 Dae-Hee Ryu By Lemma 3.4 i Fu ad Zhag 27) we have Obviously, we have, for the secod part II, II log ) δ P N ɛ log ) + +2 I =. 2.8) log ) δ P S ɛ log ) = III + IV. 2.9) Notice that Hɛ) 1 Hɛ) for M > 4 ad < ɛ < 1/4, a easy calculatio leads to uiformly with respect to < ɛ < 1/4. For IV, by Lemma 3 we have ɛ 2δ+2 III ɛ 2δ+2 log ) δ P N ɛ log ) C y 2δ+1 P N > y) as M, 2.1) M 2 lim lim M ɛ2δ+2 IV lim lim M ɛ2δ+2 log ) δ P max S k ɛ ) log =. 2.11) 1 k See Lemma 3.7 i Fu ad Zhag 27).) From 2.7) 2.11), 2.6) follows. 3. Results Propositio 4. Set Hɛ) = [expm/ɛ 2 )] ad let {X, 1} be a sequece of idetically distributed NA radom variables. If EX 2 1 log+ X 1 ) δ < for ay < δ 1. The we obtai log ) δ 1 2 ɛ log 2xP S x)dx 2xP N x ) dx =. 3.1) ɛ log Proof: Deote = sup x P S x) P N x). Assume that x = y + ɛ) log. By itegral formula ad trasformatio, it is eough to show that C 2 log ) δ 1 ɛ log 1 log ) δ 2xP S x) dx 2xP N x ) dx ɛ log 2y + ɛ) P S y + ɛ) log ) P N y + ɛ) log )

5 Precise Rates i Complete Momet Covergece for NA Sequeces 845 C 1 log ) δ [ 1/ log 1 4 2y + ɛ)p { N y + ɛ) log } 1 1/ log 4 + 2y + ɛ) P { S y + ɛ) log } P { N y + ɛ) log } + 2y + ɛ)p { S 1/ log 1 y + ɛ) log } ] 4 log )δ =: C Λ 1 + Λ 2 + Λ 3 ). 3.2) The estimates of Λ 1 ad Λ 2 are similar to those of Propositio 5.2 i Liu ad Li 26), so we omit them. It remais to estimate Λ 3, takig θ = 1/EX1 2, u = y + ɛ) log, v = ay + ɛ) log ad a = 1/{12δ + 2)} i Lemma 2.4, which yields 1 log ) δ Λ 3 C + C + C log ) δ 1/ log 1 4 4y + ɛ)p { X 1 > ay + ɛ) log } 1 log ) δ 8y + ɛ) exp 1/ log 1 4 { 1 log ) δ 8y + ɛ) 1/ log 1 4 { θ2 y + ɛ) 2 } log 8 /θ 2 4ay + ɛ) 2 log + /θ 2 } 1 12a =: Λ 4 + Λ 5 + Λ ) Note that {log Hɛ)} δ = M δ /ɛ 2δ ad EX1 2IX 1 ) as. By Toeplitz s lemma, it follows that ) ɛ 2δ Λ 4 ɛ 2δ Clog ) δ X1 E 4y + ɛ)i log y + ɛ 1/ log 1 4 a Cɛ 2δ ) CM δ 1 log Hɛ)) δ Observe that exp θ 2 /8 1/2 ɛ 2δ Λ 5 ɛ 2δ E X 1 2 I X 1 ) log ) δ 1 ) M δ 1 log Hɛ)) δ C a 2 E X 1 2 I X 1 ) log ) δ 1 as ɛ. 3.4) a 2 ) as. Usig Toeplitz s lemma agai, we have ) 32 C 1 log ) δ 1 exp θ 2 θ ) exp θ2 log ) δ as ɛ. 3.5)

6 846 Dae-Hee Ryu Observe that δ+1)/2 as. Usig Toeplitz s lemma agai, we have { θ ɛ 2δ Λ 6 ɛ 2δ C 1 log ) δ 2 y + ɛ) 2 } δ+2) log 8y + ɛ) 1/ log 1 4 3δ + 2) { ɛ 2δ C 1 log ) δ θ 2 } δ+2) 3δ + 2) 8y + ɛ) { y + ɛ) 2 log } δ+2) 1/ log 4 1 ) CM δ 1 δ+1 2 log ) δ 1 log Hɛ) δ as ɛ. 3.6) Combiig 3.2) 3.6), cosequetly, we obtai 3.1). Propositio 5. Let {X, 1} be a sequece of idetically distributed NA radom variables. If EX1 2log+ X 1 ) δ < for ay < δ 1. The we obtai, we have log ) δ 1 2xP N x ) dx =, 3.7) 2 log ) δ 1 2 ɛ log ɛ log 2xP S x)dx =. 3.8) Proof: The proof of 3.7) is quite routie, we omit it. Applyig Lemma 2.4, the proof of 3.8) are exposed as follows: For a = 1/{12δ + 2)}, we have C C 2 log ) δ 1 ɛ log 2xP{ S x}dx 1 log ) δ 2x + ɛ)p { S x + ɛ) log } dx 1 log ) δ [ {2P X1 2x + ɛ) > ax + ɛ) log } { θ 2 x + ɛ) 2 } log + 4 exp + 4 { 4θ 2 ax + ɛ) 2 log } ] 1 12a dx 8 =: C 1 log ) δ 2x + ɛ)ii 1 + II 2 + II 3 )dx. 3.9) Recallig the momet coditio, it suffices to prove that 1 log ) δ 2x + ɛ)ii 1 dx C CE log ) δ E ɛ X 2 1 x ɛ 4xI X 1 ax log ) dx log X 1 log x δ 1 I X1 x ) I X 1 M ) dx

7 Precise Rates i Complete Momet Covergece for NA Sequeces 847 Hece, for II 1, we have CEX 2 1 log X 1 log ɛ δ I X1 M ) CEX 2 1 log X 1 ) δ + CEX 2 1 log ɛ δ <. 3.1) 1 log ) δ 2x + ɛ)ii 1 dx. Next, for II 2 θ 1 log ) δ 2x + ɛ)ii 2 dx C 1 log ) δ 2 x 2 ) log 8x exp dx ɛ 8 ) C 1 log ) δ 32 exp θ2 ɛ 2 ) log θ 2 log 8 ) 32 C 1 θ2 ɛ 2 θ 2 8 <. 3.11) Hece 1 log ) δ 2x + ɛ)ii 2 dx as ɛ. 3.12) Fially, for II 3 { θ 1 log ) δ 2x + ɛ)ii 3 dx C 1 log ) δ 2 x 2 } δ+2) log 8x dx ɛ 3δ + 2) ) { C 1 log ) 2 4 θ 2 } δ+2) ɛ 2δ ) δ + 1 3δ + 2) Hece 1 log ) δ 2x + ɛ)ii 3 dx Cɛ 2 4 δ + 1 Cɛ 2 uiformly with respect to ɛ. Hɛ) ) δ ) δ+2) Hɛ) 1 log ) 2 dx xlog x) 2 CM 1 as M, Theorem 1. Let {X, 1} be a sequece of idetically distributed NA radom variables. EX1 2log+ X 1 ) δ < for ay < δ 1, the 1.4) holds. Proof: I fact, oe ca easily get log ) δ 1 ES I 2 S 2 ɛ log ) =1 = ɛ 2 log ) δ P S ɛ log ) log ) δ 1 + =1 =1 2 ɛ log 2xP S x) dx =: I 1 + I ) If

8 848 Dae-Hee Ryu Cosequetly to verity Theorem 1 we oly eed to cosider I 1 ad I 2, respectively. From Propositios 1 ad 3 we have I 1 = E N 2δ+2 δ ) It follows from Propositios 4 ad 5 that log ) δ 1 2 2xP S x) dx 2xP N ) x dx =. 3.16) ɛ log ɛ log =1 Hece, by Propositio 2 ad 3.16) we also have I 2 = E N 2δ+2 δδ + 1), 3.17) which completes the proof together with 3.15). Next we will show that if S is replaced by M = max 1 k S k, the Theorem 1 still holds. Propositio 6. Fu ad Zhag, 27) Suppose that {Wt); t } is a stadard Wieer processbrowiia motio). The, for ay < δ 1, +2 log ) δ =1 Proof: Refer to Huag ad Zhag 25). P sup Ws) ɛ ) log = 2E N 2δ+2 δ 1 δ + 1 = 1) 2 + 1) 2δ+2. Propositio 7. Zhao, 28) Suppose that {Wt); t } is a stadard Wieer process. The, for ay < δ 1, log ) δ 1 =2 2 ɛ log 2xP sup Wt) x ) = 2E N 2δ+2 t 1 δδ + 1) = 1) 2 + 1) 2δ+2. Theorem 2. Let {X, 1} be a sequece of idetically distributed NA radom variables. EX1 2log+ X 1 ) δ < for ay < δ 1, the 1.5) holds. Proof: Note that Theorem 2 is the maximal versio of Theorem 1. Hece, if we make some modificatio of the proof of Theorem 1, Theorem 2 will follow. As i 3.14), ideed, it suffices to stu that log ) δ 1 EM I 2 M 2 ɛ log ) =2 = ɛ 2 log ) δ P M ɛ log ) log ) δ 1 + =2 =: I 3 + I 4. =2 2 ɛ log 2xP M x) dx To pave the way for the proofs of I 3 of I 4 alog the same lies as that of the proof of Theorem 1, together with Lemmas 2 ad 3 ad Propositios 6 ad 7. If

9 Precise Rates i Complete Momet Covergece for NA Sequeces Cocludig Remark It is of iterest to show that the precise rate result i this kid of complete momet covergece also holds for movig average process. I the future stu, we ivestigate the precise rate of covergece i complete momet of movig average processes based NA radom variables by extedig Theorems 1 ad 2 to the movig average processes. Refereces Alam, K. ad Saxea, K. M. L. 1981). Positive depedece i multivariate distributios, Commuicatios i Statistics - Theory ad Methods, 1, Joag-Dev, K. ad Proscha, F. 1983). Negative associatio of radom variables with applicatios, The Aals of Statistcis, 11, Fu, K. A. ad Zhag, L. X. 27). Precise rates i the law of the logarithm for egatively associated radom variables, Computers & Mathematics with Applicatios, 54, Huag, W. ad Zhag, L. X. 25). Precise rates i the law of the logarithm i the Hilbert space, Joural of Mathematical Aalysis ad Applicatios, 34, Kim, T. S., Lee, S. W. ad Ko, M. H. 21). O the estimatio of empirical distributio fuctio for egatively associated processes, The Korea Commuicatio i Statistics, 8, Ko, M. H. 29). A cetral limit theorem for the liear process i a Hilbert space uder egative associatio, Commuicatio of the Korea Statistical Society, 16, Li, Y. X. ad Zhag, L. X. 24). Complete momet covergece of movig- average processes uder depedece assumptios, Statistics & Probability Letters, 7, Liag, H. Y. 2). Complete covergece for weighted sums of egatively associated radom variables, Statistics & Probability Letters, 48, Liu, W. D. ad Li, Z. Y. 26). Precise asymptotics for a ew kid of complete momet covergece, Statistics & Probability Letters, 76, Newma, C. M. 1984). Asymptotic idepedece ad limit theorems for positively ad egatively depedet radom variables, Y. L. Tog, ed. Iequalities i Statistics ad Probability: Proceedigs of the Symposium o Iequalities i Statistics ad Probability, 5, Peligrad, M. 1987). The covergece of momets i the cetral limit theorem for ρ-mixig sequece of radom variables, I Proceedigs of the America Mathematical Society, 11, Shao, Q. M. ad Su, C. 1999). The law of the iterated logarithm for egatively associated radom variables, Stochastic Processes ad their Applicatios, 83, Shao, Q. M. 2). A compariso theorem o momet iequalities betwee egatively associated ad idepedet radom variables, Joural of Theoretical Probability, 13, Zhao, Y. 28). Precise rates i complete momet covergece for ρ-mixig sequeces, Joural of Mathematical Aalysis ad Applicatios, 339, Received July 29; Accepted August 29

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