Complete Convergence for Asymptotically Almost Negatively Associated Random Variables

Transcription

1 Applied Mathematical Scieces, Vol. 12, 2018, o. 30, HIKARI Ltd, Complete Covergece for Asymptotically Almost Negatively Associated Radom Variables Cog Qu, Shuili Zhag ad Tiatia Hou College of Mathematics ad Statistics Pigdigsha Uiversity, Hea Provice, , Chia Copyright c 2018 Cog Qu, ShuiLi Zhag ad Tiatia Hou. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract I this paper, some sufficiet coditios of complete covergece for asymptotically almost egatively associated radom variables are obtaied, which partially exted the correspodig oes for idepedet radom variables ad egatively associated radom variables. Mathematics Subject Classificatio: 60F15 Keywords: Complete covergece, asymptotically almost egatively associated, stochastically domiated 1. Itroductio 1.1. Complete covergece. A sequece of radom variables {U, 1} is said to coverge completely to a costat C if P U C > ε <, for all ε > The cocept of complete covergece was itroduced firstly by Hsu ad Robbis [1]. I view of the Borel-Catelli lemma, complete covergece implies that U C almost surely. The coverse is true if {U, 1} are idepedet radom variables. Hsu ad Robbis [1] proved that the sequece of arithmetic meas of idepedet ad idetically distributed i.i.d. radom

2 1442 Cog Qu, ShuiLi Zhag ad Tiatia Hou variables coverges completely to the expected value if the variace of the summads is fiite. Somewhat later, Erdös [2] proved the coverse. We summarize their results as follow. Hsu-Robbis-Erdös Strog Law. Let {X, X, 1} be a sequece of i.i.d. radom variables with mea zero, ad set S = X i, 1, the EX 2 < is equivalet to the coditio that P S > ε <, for all ε > Hsu-Robbis-Erdös strog law ca be viewed as a result o the rate of covergece i the law of large umbers. The followig theorem is a more geeral result which bridges the itegrability of summads ad the rate of covergece i the Marcikiewicz-Zygmud strog law of large umbers. Theorem A. Let 0 < r < 2, r p. Suppose that {X, X, 1} is a sequece of i.i.d. radom variables with mea zero, ad set S = X i, 1, the E X p < is equivalet to the coditio that p/r 2 P S > ε 1/r <, for all ε > 0, 1.3 ad also equivalet to the coditio that p/r 2 S k > ε 1/r <, for all ε > k For r = p = 1, the equivalece betwee E X < ad 1.3 is a famous result due to Spitzer [3]. For p = 2 ad r = 1, the equivalece betwee EX 2 < ad 1.3 is just Hsu-Robbis-Erdös strog law. For the geeral p, r satisfyig the coditios of Theorem A, Katz [4], ad later Baum ad Katz [5] proved the equivalece betwee E X p < ad 1.3 ad Chow [6] established the equivalece betwee E X p < ad 1.4. For the i.i.d. case, related results are fruitful ad detailed. It is atural to exted them to depedet cases, for examples, martigale differece, egatively associated sequece, mixig radom variables ad so o. I the preset paper, we are iterested i the asymptotically almost egatively associated radom variables Some otatios. First, let us recall some defiitios as follow. Defiitio 1.1 The radom variables {X, 1} is stochastically domiated by the radom variable X if there exists a positive costat C such that, for all x > 0 ad 1. P X > x P X > x 1.5

3 Complete covergece for AANA radom variable 1443 Defiitio 1.2 [7] A fiite family of radom variables {X k, 1 k } is said to be egatively associated NA, if for ay disjoit subsets A ad B of {1, 2,, } ad ay real coordiatewise odecreasig fuctios f o R A ad g o R B, CovfX i, i A, gx j, j B 0, wheever the covariace exists. A ifiite family of radom variables is NA if every fiite subfamily is NA. Defiitio 1.3[8] A sequece {X, 1} of radom variables is said to be asymptotically almost egatively associatedaana if there exists a oegative sequece q 0 as such that CovfX, gx +1, X +2,, X +k q [V arfx V argx +1, X +2,, X +k ] 1/2 1.6 for all, k 1 ad all coordiate-wise odecreasig cotiuous fuctios f ad g wheever the variaces exist. {q, 1} is called the mixig coefficiets of {X, 1}. Obviously, the family of AANA sequeces cotai NA sequecewith q = 0, 1. A example of AANA radom variables which are ot NA was costructed by Chadra ad Ghosal[8]. Sice the otio of AANA sequece was itroduced by Chadra ad Ghosal[8], the AANA properties have aroused wide iterest because of umerous applicatios i reliability theory ad multivariate statistical aalysis. The probability iequality, momet iequality, probability limit theorems ad applicatio for AANA radom variables were obtaied. For more details o related results, oe ca refer to Chadra [8], Yua[9], Wag[10, 11, 12], Wag[13], A[14], Ko[15], Tag[16], ad so o. Throughout the paper, C deote positive costats which may be differet i variables places. IA be the idicator fuctios of the set A. The a = Ob deote that there exists a positive costat C such that a b. 2. Mai results I the sectio, we state our mai results. Theorem 2.1. Let {X, 1} be a sequece of AANA radom variables with mixig coefficiets {q, 1}, ad q2 <, which is stochastically domiated by X. Let {a i, 1 i, 1} be a array of costats such that for some 0 < α < 2, a i α = O, 2.1 Let b = 1 α log 1 α, if EX = 0 for 1 < α < 2 ad E X α log1+ X <, the for ay ɛ > 0,

4 1444 Cog Qu, ShuiLi Zhag ad Tiatia Hou 1 P a i X i > ɛb <. 2.2 Theorem 2.2. Let {X, 1} be a sequece of AANA radom variables with mixig coefficiets {q, 1}, ad q2 <. Let 1 p 2, if p E X p <, 2.3 ad satisfy the Cesàro uiform itegrality coditio, i.e., lim 1 E X k I X k > x = The for ay ɛ > 0, sup x 1 I particular, 1 k X k EX k 1 k > ɛ < X k EX k 0, a.s Proofs of Mai results 3.1. Some lemmas. To prove our results, we first give some lemmas as follows. Lemma 3.1. [9] Let {X, 1} be a sequece of AANA radom variables with mixig coefficiets {q, 1}. Let f 1, f 2, be all odecreasig or all oicreasig fuctios, the {f X, 1} is still a sequece of AANA radom variables with mixig coefficiets {q, 1}. Lemma 3.2. [9] Let {X, 1} be a AANA sequece of zero mea radom variables with mixig coefficiets {q, 1}. If q2 <, the there exists a positive costat C p depedig oly o p such that k p E max X i p E X i p k for all 1 ad 1 < p 2. Lemma 3.3. [17] Let X be a radom variables, {a i, 1 i, 1} be a array of positive costats such that 2.1 for some 0 < α < 2. Let

5 Complete covergece for AANA radom variable 1445 b = 1/α log 1/γ for some γ > 0. The CE X α, 1 P a i X > b E X α log1 + X, CE X γ, if α > γ if α = γ if α < γ 3.2 Lemma 3.4. [17] Let X be a radom variables, b = 1/α log 1/α, {a i, 1 i, 1} be a array of positive costats such thata i = 0 or a i > 1, ad a i α for some α > 0.If q > α, the 1 b q E a i q I a i X b E X α log1 + X. 3.3 Lemma 3.5. [18] Let {X, 1} be a sequece of radom variables which is stochastically domiated by a radom variable X. If E X p < for some p > 0, the for ay t > 0 ad 1, the followig statemets hold: 1 E X k p E X p, 3.4 ad 1 E X k p I X k t [E X p I X t + t p P X > t] E X k p I X k > t E X p I X > t. 3.6 Lemma 3.6. Let 1 p 2, {X, 1} be a sequece of AANA radom variables with mixig coefficiets {q, 1}, ad q2 <. If p E X p <, the for ay ε > 0, 1 P X k EX k > ε <. 3.7 Proof. By p E X p < ad Kroecker lemma, we ca get that E X k p 0, as. 3.8 p Hece for ay ε > 0, we have that for large eough 1 EX k I X k > 1 E X k I X k > 1 1 p E X k p I X k > p E X k p ε

6 1446 Cog Qu, ShuiLi Zhag ad Tiatia Hou By Lemma 3.2, {X k I X k EX k I X k, 1, 1 k } is a sequece of AANA radom variables with zero mea. From3.9, Lemma 3.3 ad Markov s iequality, we ca get that 1 P X k EX k > ε 1 P X k > + 1 P X k IX k EX k IX k > ε 2 1 p E X k p + C 3 E X k 2 I X k 3.10 C p 1 E X k p E X k p + C =k p 1 p 1 E X k p k p E X k p < Proof of Theorem 2.1. With loss of geerality, we may assume that a i α, ad a i 0 for all 1 i ad 1. For 1, ad 1 i, let ad X i = b Ia i X i < b + a i X i I a i X i b + b Ia i X i > b, Z i = a i X i + b Ia i X i < b + a i X i b Ia i X i > b. It is easy to see that for 1, 1 i, we have X i + Z i = a i X i. The 1 P a i X i > ɛb 1 P { a i X i > b } + 1 P a i X i > b + 1 P X i > ɛb, { a i X i b } 1 P X i > ɛb I 1 + I Therefore, to prove 2.2, it is eough to show that I 1 < ad I 2 <. By the Lemma 3.3 ad the coditio E X α log1 + X <, we have I 1 E X α log1 + X <. 3.12

7 Complete covergece for AANA radom variable 1447 To prove I 2 <, we first show that b 1 EX i 0, For 1 < α < 2, ote that EX i = 0 ad X i +Z i = a i X i, the EX i = EZ i. By Markov s iequality ad Lemma 3.5, we have b 1 b α EX i = b 1 P a i X i > b + b 1 EZ i b 1 E Z i E a i X i I a i X i > b a i α E X α + Cb α a i α E X α E X α log 1 0, as. For 0 < α 1, by Lemma 3.5 ad Markov s iequality, the b 1 EX i b 1 E X i P a i X i > b + b 1 b α E a i X i I a i X i b P a i X > b + Cb 1 P a i X > b + Cb 1 a i α E X α + Cb α 3.14 [ E a i X I a i X b + b P a i X > b ] E a i X I a i X b a i α E X α E X α log 1 0, as. Hece, to prove I 2 <, it is eough to prove that I 3 = 3.15 P X i EX i > ɛb <, for all ɛ > From Lemma 3.1, it is easy to see that for ay fixed 1, {X i EX i, 1 i } are AANA radom variables with mea zero. By Markov s iequality

8 1448 Cog Qu, ShuiLi Zhag ad Tiatia Hou ad Lemma 3.2, we get that I 3 = 1 E X i EX i > ɛb 1 b 2 EX i EX i 2 1 b 2 1 b 2 1 b 2 1 b 2 E 2 X i EX i 1 b 2 EX 2 i [ Ea i X i 2 I a i X i b + b 2 P a i X i > b ] [ Ea i X 2 I a i X b + b 2 P a i X > b ] Ea i X 2 I a i X b + C 1 P a i X > b I 4 + I Similar to I 1 <, we have I 5 <. To prove I 4 <, we divide {a i, 1 i } ito three subsets {a i : a i 1/log s }, {a i : 1/log s < a i 1}, {a i : a i > 1}, where s = 1/2 α.the I 4 = = b 2 1 b 2 1 b 2 1 b 2 Ea i X 2 I a i X b i: a i 1/log s Ea i X 2 I a i X b i:1/log s < a i 1 i: a i >1 I 41 + I 42 + I 43. Ea i X 2 I a i X b Ea i X 2 I a i X b 3.18 By Lemma 3.4, we get that I 43 <. Note that i: a i 1/log s a i α log sα,

9 ad E X α <, the Complete covergece for AANA radom variable 1449 I 41 = 1 b 2 1 b α 1 b α i: a i 1/log s Ea i X 2 I a i X b i: a i 1/log s Ea i X α I a i X b i: a i 1/log s a i α 1 log 1 sα < Note that i:1/log s < a i 1 a i 2, ad E X α <, the I 42 = = C 1 b 2 1 b 2 i:1/log s < a i 1 i:1/log s < a i 1 b 2 EX 2 I X 1/α log s+1/α b 2 Ea i X 2 I a i X b a 2 iex 2 I X b log s EX 2 I i 1 1/α logi 1 s+1/α < X i 1/α log i s+1/α EX 2 I i 1 1/α logi 1 s+1/α < X i 1/α log i s+1/α 2/α log 2/α =i =i EX 2 I i 1 1/α logi 1 s+1/α < X i 1/α log i s+1/α log i 2/α 2/α EX 2 I i 1 1/α logi 1 s+1/α < X i 1/α log i s+1/α log i 2/α 1 2/α E X α <. 3.20

10 1450 Cog Qu, ShuiLi Zhag ad Tiatia Hou 3.3. Proof of Theorem 2.2. For give ε > 0, by the Cesàro uiform itegrality coditio, there is a costat x = xε > 0, such that for all 1, 1 EX k I X k > x 1 E X k I X k > x ε For x = xε, we have 1 k X i EX i 1 k > ɛ 1 k X i I X i x EX i I X i x 1 k > ɛ k X i I X i > x EX i I X i > x 1 k > 3ɛ 4 J 1 + J To prove 2.5, it is eough to prove J 1 < ad J 2 <. By Markov s iequality ad Lemma 3.2, J 1 = 1 k X i I X i x EX i I X i x 1 k > ɛ 4 3 E k 2 max X i I X i x EX i I X i x 1 k 3 EXi 2 I X i x 2 <. 3.23

11 Complete covergece for AANA radom variable 1451 By 3.21, Markov s iequality ad Lemma 3.6, the J 2 = 1 k X i I X i > x EX i I X i > x 1 k > 3ɛ 4 1 k X i I X i > x 1 k > ɛ 2 1 P X i I X i > x > ɛ 2 1 P X i I X i > x E X i I X i > x > ɛ < Refereces [1] P.L. Hsu, H. Robbis, Complete covergece ad the law of large umber, Proc. Natl. Acad. Sci., , [2] P. Erdös, O a theorem of Hsu ad Robbis, A. Math. Statist., , [3] F. Spitzer, A combiatorial lemma ad its applicatios to probability theory, Tras. Amer. Math. Soc., , [4] M. Katz, The probability i the tail of a distributio, A. Math. Statist., , [5] L.E. Baum, M. Katz, Covergece rates i the law of large umbers, Tras. Amer. Math. Soc., , [6] Y.S. Chow, Delayed sums ad Borel summability of idepedet, idetically distributed radom variables, Bull. Ist. Math. Acad. Siica, , [7] K. Alam, K.M.L. Saxea, Positive depedece i multivariate distributios, Comm. Statist-Theory ad Methods, , [8] T.K. Chadra, S. Ghosal, The strog law of large umber for weighted averages uder depedet assumptios, J. Theor. Probab., , [9] D.M. Yua, J. A, Rosethal type iequalities for asymptotically almost egatively associated radom variables ad applicatios, Sci. Chia Ser. A: Math., , [10] X.J. Wag, S.H. Hu, W.Z. Yag, Covergece properties for asymptotically almost egatively associated sequece, Discrete. Dy. Nat. Soc., , 1-15, Article ID [11] X.J. Wag, S.H. Hu, W.Z. Yag, Maximal iequalities ad strog law of large umber umber for AANA sequeces, Commu. Korea. Math. Soc., , [12] X.J. Wag, S.H. Hu, W.Z. Yag, Complete covergece for arrays asymptotically almost egatively associated radom variables, Discrete. Dy. Nat. Soc., , 1-11, Article ID

12 1452 Cog Qu, ShuiLi Zhag ad Tiatia Hou [13] X.H. Wag, A.T. She, X.Q. Li, A ote o complete covergece of weighted sums for arrays of rowwise AANA radom variables, J. Iequal. Appl., , [14] J. A, Complete momet covergece of weighted sums for processes uder asymptotically almost egatively associated assumptips, Proc. Idia. Acad. Sci., , [15] M.H. Ko, D.H. Ryu, T.S. Kim, The almost sure covergece of AANA sequece of AANA sequeces i double arrays, Bull. Korea. Math. Soc., , [16] X.F. Tag, Some strog laws of large umbers for weighted sums of asymptotically almost egatively associated radom variables, J. Iequal. Appl., , 4. [17] S.H. Sug, O the strog covergece for weighted sums of ρ mixig radom variables, Stat. Pap., , [18] A. Kuzmaszewska, O complete covergece i Marcikiewicz-Zygmud type SLLN for egatively associated radom variables, Acta. Math. Hugar., , Received: October 19, 2018; Published: November 28, 2018