Condtonal lmt theorems for condtonally negatvely assocated random varables Monatshefte f?r Mathematk ISSN 006-955 Volume 161 Number 4 Monatsh Math 010 161:449-473 DOI 10.1007/s00605-010-0196- x 1 3
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Monatsh Math 010 161:449 473 DOI 10.1007/s00605-010-0196-x Condtonal lmt theorems for condtonally negatvely assocated random varables De-Me Yuan Jun An Xu-Shan Wu Receved: Aprl 009 / Accepted: 19 January 010 / Publshed onlne: 9 February 010 Sprnger-Verlag 010 Abstract From the ordnary noton of negatve assocaton for a sequence of random varables, a new concept called condtonal negatve assocaton s ntroduced. The relaton between negatve assocaton and condtonal negatve assocaton s answered, that s, the negatve assocaton does not mply the condtonal negatve assocaton, and vce versa. The basc propertes of condtonal negatve assocaton are developed, whch extend the correspondng ones under the non-condtonng setup. By means of these propertes, some Rosenthal type nequaltes for mum partal sums of such sequences of random varables are derved, whch extend the correspondng results for negatvely assocated random varables. As applcatons of these nequaltes, some condtonal mean convergence theorems, condtonally complete convergence results and a condtonal central lmt theorem stated n terms of condtonal characterstc functons are establshed. In addton, some lemmas n the context are of ndependent nterest. Keywords Condtonal negatve assocaton Condtonal Rosenthal type nequalty Condtonal statonarty Condtonal mean convergence theorem Condtonally complete convergence Condtonal central lmt theorem Condtonal resdual Cesàro alpha-ntegrablty Condtonal strong resdual Cesàro alpha-ntegrablty Communcated by K.D. Elworthy. D.-M. Yuan B J. An X.-S. Wu College of Mathematcs and Statstcs, Chongqng Technology and Busness Unversty, Chongqng 400067, Chna e-mal: yuandeme@163.com J. An e-mal: scottan@sna.com X.-S. Wu e-mal: xushan191@163.com
450 D.-M. Yuan et al. Mathematcs Subject Classfcaton 000 60F15 60F5 1 Introducton, defnton and basc propertes Let, A, P be a probablty space, and all random varables n ths paper are defned on t unless specfed otherwse. A fnte famly of random varables {X, 1 n} s sad to be negatvely assocated NA, n short f for every par of dsjont subsets A and B of {1,,...,n}, Cov f 1 X, A, f X j, j B 0 whenever f 1 and f are coordnatewse nondecreasng and the covarance exsts. An nfnte famly s NA f every fnte subfamly s NA. Ths concept was frst ntroduced by Alam and Saxena [1]. Joag-Dev and Proschan [7] showed that many well known multvarate dstrbutons possess the NA property, and ts lmt propertes have drawn wde attenton because of ther numerous applcatons n relablty theory, percolaton theory and multvarate statstcal analyss. We refer to Joag-Dev and Proschan [7] for fundamental propertes, Matula [1] for the three seres theorem, Su et al. [18] and Shao [17] for moment nequaltes, Shao and Su [16] for the law of the terated logarthm, Lang and Su [9] for complete convergence, Newman [14] for the central lmt theorem, Ln [10] for the nvarance prncple, among others. Let X and Y be random varables wth EX < and EY <. LetF be a sub-σ -algebra of A. Prakasa Rao [15] defned the noton of the condtonal covarance of X and Y gven F F-covarance, n short as [ ] Cov F X, Y = E F X E F XY E F Y, where E F Z denotes the condtonal expectaton of a random varable Z gven F. It s easy to see that the F-covarance reduces to the ordnary concept of covarance f F ={Ø, }. We pont out that Cov F X + Z, Y + W = Cov F X, Y and Cov F XZ, YW = ZWCov F X, Y are two easy consequences of the above defnton of condtonal covarance, provded that Z and W are F-measurable random varables. On the bass of the above defnton of condtonal covarance, we now consder a new knd of dependence called condtonal negatve assocaton, whch s an extenson to the correspondng non-condtonal case. Defnton 1 A fnte famly of random varables {X, 1 n} s sad to be condtonal negatvely assocated gven F F-NA, n short f, for every par of dsjont subsets A and B of {1,,...,n}, Cov F f 1 X, A, f X j, j B 0 a.s. whenever f 1 and f are coordnatewse nondecreasng and the F-covarance exsts. An nfnte famly s F-NA f every fnte subfamly s F-NA.
Condtonal lmt theorems for condtonally negatvely assocated random varables 451 Clearly, a sequence of random varables {X n, n 1} s F-NA f they are F-ndependent. Moreover, the F-negatve assocaton reduces to the concept of negatve assocaton when F ={Ø, }. The natural queston of the relaton between the two concepts of NA and condtonal NA arses. The followng examples show that the NA property of random varables may be nherted by the condtonal NA property, that the NA property does not mply the condtonal NA property and that the condtonal NA property does not mply the NA property. Example 1 Consder the random vector X 1, X, X 3 whch has the multnomal dstrbuton wth parameters n = 3 and p = 1/3, 1/3, 1/3. As t s well known, the random vector X 1, X has also the multnomal dstrbuton. It follows that the famly {X 1, X } s NA n vew of Joag-Dev and Proschan [7]. Now let B ={X 3 = } and let F ={, B, B c, Ø} be the sub-σ -algebra generated by the event B, we next prove that the famly {X 1, X } s F-NA. For every par of functons f 1 and f wth Ef 1 X 1 and Ef X fnte, some smple calculatons show that and E[ f 1 X 1 B], ω B, E F f 1 X 1 = E[ f 1 X 1 B c ], ω B c = E[ f X B], ω B, E F f X = E[ f X B c ], ω B c = 1 f 10 + 1 f 11, ω B, 5 1 f 10 + 3 7 f 11 + 7 f 1 + 1 1 f 13, ω B c, 1 f 0 + 1 f 1, ω B, 5 1 f 0 + 3 7 f 1 + 7 f + 1 1 f 3, ω B c E[ f 1 X 1 f X B], ω B, E F [ f 1 X 1 f X ]= E[ f 1 X 1 f X B c ], ω B c 1 f 10 f 1 + 1 f 11 f 0, ω B, 1 1 f 10 f 0 + 1 7 f 10 f + 1 1 f 10 f 3 = + 7 f 11 f 1 + 1 7 f 11 f + 1 7 f 1 f 0 + 1 7 f 1 f 1 + 1 1 f 13 f 0, ω B c.
45 D.-M. Yuan et al. If f 1 and f are nondecreasng, then and 0 f 1 1 f 1 0 f 1 f 0 3 f 1 1 f 1 0 f 1 f 0 33 f 1 1 f 1 0 f f 1 + 9 f 1 1 f 1 0 f 3 f 1 + 36 f 1 f 1 1 f f 0 + 6 f 1 3 f 1 1 f f 0 + f 1 3 f 1 0 f f 0 + 6 f 1 f 1 0 f 3 f 0 + 9 f 1 3 f 1 f 1 f 0 + 6 f 1 3 f 1 1 f f 0 + f 1 3 f 1 0 f 3 f 0, equvalently, E F [ f 1 X 1 f X ] E F f 1 X 1 E F f X a.s., whch shows that the famly {X 1, X } s F-NA. Remark Let the random vector X 1,...,X r, X r+1,...,x m have multnomal dstrbuton wth parameters n and p 1,...,p m, and let F = σx r+1,...,x m.we conjecture that the famly {X, 1 r} s F-NA. Example Let ={1,, 3, 4} and let p = 1/4 be the probablty assgned to the event {}. If the events A 1 and A are defned by A 1 ={1, } and A ={, 3}, and the random varables X 1 and X are as follows X 1 = I A1 and X = I A where I A denotes the ndcator functon of an event A, then X 1 and X are ndependent, so the famly {X 1, X } s NA. Let B ={1} and F ={, B, B c, Ø} be the sub-σ -algebra generated by B. We wll show that {X 1, X } s not F-NA. For any nondecreasng functons f 1 and f wth Ef 1 X 1 and Ef X fnte and satsfyng f 1 0 = f 0 = 1 and f 1 1 = f 1 =, some smple calculatons show that f 1 1, ω B,, ω B, E F f 1 X 1 = = 3 f 1 0 + 1 3 f 11, ω B c 4 3, ω B c, f 0, ω B, 1, ω B, E F f X = = 1 3 f 0 + 3 f 1, ω B c 5, 3, ω Bc and { E F f 1 1 f 0, ω B, [ f 1 X 1 f X ]= 1 3 [ f 10 f 0 + f 1 0 f 1 + f 1 1 f 1], ω B c {, ω B, = 7 3, ω Bc.
Condtonal lmt theorems for condtonally negatvely assocated random varables 453 Hence E F [ f 1 X 1 f X ] E F f 1 X 1 E F f X, ths shows that the famly {X 1, X } s not F-NA. Example 3 Let ={1,, 3, 4, 5, 6} and let p = 1/6 be the probablty assgned to the event {}. Defne the events A 1 and A by A 1 ={1,, 3, 4} and A ={, 3, 4, 5}, and the random varables X 1 and X by X 1 = I A1 and X = I A.LetB ={6} and let F ={, B, B c, Ø} be the sub-σ -algebra generated by the event B. For any strctly ncreasng functons f 1 and f wth Ef 1 X 1 and Ef X fnte, some smple calculatons show that f 1 0, ω B, E F f 1 X 1 = 1 5 [ f 10 + 4 f 1 1], ω B c, f 0, ω B, E F f X = 1 5 [ f 0 + 4 f 1], ω B c and f 1 0 f 0, ω B, E F [ f 1 X 1 f X ]= 1 5 [ f 10 f 1 + f 1 1 f 0 + 3 f 1 1 f 1], ω B c. Note that Hence f 1 0[ f 1 f 0] f 1 1[ f 1 f 0]. E F [ f 1 X 1 f X ] E F f 1 X 1 E F f X, ths shows that the famly {X 1, X } s F-NA. But the famly {X 1, X } s not NA as follows from the observaton E[ f 1 X 1 f X ]= 1 6 [ f 10 f 0 + f 1 0 f 1 + f 1 1 f 0 + 3 f 1 1 f 1] 1 3 f 10 + 1 3 f 11 3 f 0 + 3 f 1 = Ef 1 X 1 Ef X. A concrete example where condtonal lmt theorems are useful s the study of statstcal nference for non-ergodc models as dscussed n Basawa and Prakasa Rao [] and Basawa and Scott [3]. For nstance, f one wants to estmate the mean off-sprng
454 D.-M. Yuan et al. θ for a Galton-Watson Branchng process, the asymptotc propertes of the mum lkelhood estmator depend on the set of non-extncton. As t was ponted out earler, the condtonal NA property does not mply the NA property and the opposte mplcaton s also not true. Hence one does have to derve lmt theorems under condtonng f there s a need for such results even through the proofs of such results may be analogous to those under the non-condtonng setup. Consequently, the study of the lmt theorems for sequences of condtonal NA random varables s of much nterest. Our am n ths paper s to dscuss condtonal lmt theorems for F-NA random varables. Precsely, we wll derve condtonal versons of several knds of lmt theorems: condtonal Rosenthal type nequalty, condtonal mean convergence theorem, condtonal complete convergence and condtonal central lmt theorem for sequences of F-NA random varables. We put S n = n X for a sequence of random varables {X n, n 1}. Let F be a sub-σ -algebra of A. {X n, n 1} wll be called condtonally centered f E F X n = 0 for every n 1, and t s called condtonally statonary gven F f for all 1 t 1 < < t k <, r 1, the jont dstrbuton of X t1,...,x tk condtoned on F s the same as the jont dstrbuton of X t1 +r,...,x tk +r condtoned on F a.s. From now on, we assume that the condtonal expectatons under consderaton exst and the condtonal dstrbutons exst as regular condtonal dstrbutons cf. Chow and Techer [5]. We conclude ths secton by some basc propertes of F-NA random varables. The frst property s mmedate from the defnton of an F-NA sequence. Proposton P1 Coordnatewse nondecreasng or nonncreasng functons defned on dsjont subset of a set of F-NA random varables are F-NA. The followng property wll be used frequently n the subsequent sectons. Proposton P If two sets of F-NA random varables are F-ndependent of one another, then ther unon s a set of F-NA random varables. In partcular, f {X n, n 1} s F-NA, then {X n E F X n, n 1} s condtonally centered and F-NA. Proof Let X = X 1, X,...,X n, Y = Y 1, Y,...,Y m be F-ndependent vectors of each other, each F-NA, Let X 1, X and Y 1, Y denote arbtrary parttons of X and Y, respectvely. Let f 1 and f be arbtrary coordnatewse nondecreasng functons wth fnte expectatons as need. Wrtng f 1 for f 1 X 1, Y 1, and f for f X, Y. Snce X and Y are F-ndependent, we have E F X,Y f 1 f = E F X EF Y f 1 f by takng f = u I A and by usng the usual approxmaton method, where E F X denotes condtonal expectaton over the dstrbuton of X, EY F denotes condtonal expectaton over the dstrbuton of Y, and E F X,Y denotes condtonal expectaton over the jont dstrbuton of X and Y.
Condtonal lmt theorems for condtonally negatvely assocated random varables 455 Snce f 1 x 1, y 1 and f x, y are nondecreasng functons n y 1 and y, respectvely, we conclude that E F Y f 1 f E F Y f 1 E F Y f, and therefore E F X,Y f 1 f E F X EY F f 1 EY F f. Note that EY F f 1x 1, Y 1 and EY F f x, Y are nondecreasng functons n x 1 and x, respectvely, so that EY F f 1 EY F f E F X EF Y f 1 E F X EF Y f. Hence E F X E F Y f 1 f E F X,Y f 1 E F X,Y f agan from the F-ndependence of X and Y. Condtonal Rosenthal type nequaltes The followng condtonal Rosenthal type nequaltes for F-NA random varables extend the correspondng results for NA random varables, see for example, Theorem of Shao [17] or Theorem 1 of Su et al. [18]. Moreover, these are the man tools for studyng the lmt results n the subsequent sectons. Theorem 1 Let {X n, n 1} be a sequence of condtonally centered F-NA random varables. Then there exsts a postve constant C p dependng only p such that for all n 1, E F S k p C p 1 k n and E F S k p C p 1 k n E F X p a.s. for 1 p 1 p/ E F X p + E F X a.s. for p >. Proof The basc approach of ths proof s based on Yuan and An [1] but the detals are qute dfferent. Wthout loss of generalty, we can assume that E F X p <
456 D.-M. Yuan et al. a.s., 1, for otherwse both the rght-hand sde of 1 and the rght-hand sde of are almost surely nfnty and there s nothng to prove. We frst prove 1. Set U = X, X + X +1,...,X + + X n, 1 n. Clearly, U = X, X + U +1. From the elementary nequalty x + y p p x p + y p + px y p 1 sgn{y} for 1 p, 3 t follows that for 1 n 1 E F U p E F X p I U +1 0 + E F X + U +1 p I U +1 > 0 p E F X p + E F U +1 p + pe F X U +1 p 1 I U +1 > 0 a.s. for 1 p. It s easy to show that gx +1,...,X n := U +1 p 1 I U +1 > 0 s a coordnatewse nondecreasng functon of X j. Thus, by the hypothess of F-NA property, we have for 1 n 1 E F U p p E F X p + E F U +1 p a.s. for 1 p. 4 Substtutng sequentally, we conclude that E F 1 k n S k p p E F X p a.s. for 1 p. 5 By P1, { X n, n 1} s also a sequence of condtonally centered F-NA random varables. In the same way, we also have E F S p k 1 k n p E F X p a.s. for 1 p. 6 Observe that 1 k n S k p 1 k n S k p + S p k 1 k n. 7 Relaton 1 follows mmedately by combnng 5, 6 and 7. Now we turn to the proof of, by nducton on p. In the same manner, wth the excepton of relaton 3 beng replaced by the followng elementary nequalty x + y p p x p + y p + px y p 1 sgn{y}+ p p x y p for p >, we obtan the analogue of 4 E F U p p E F X p + E F U +1 p + p p E F X U +1 p I U +1 > 0
Condtonal lmt theorems for condtonally negatvely assocated random varables 457 a.s. for p >. Summng over 1 n 1 on both sde of the above nequalty, we conclude that E F 1 k n S k p p E F n 1 X p + p p a.s. for p >. Observe that U p +1 I U +1 > 0 = 0, X +1, X +1 + X +,...,X +1 + + X n p E F X U +1 p I U +1 > 0 = S, S + X +1, S + X +1 + X +,...,S + X +1 + + X n S p p p + S p 1 k n S k p 1 1 k n S k p. Hence E F 1 k n S k p p E F X p + p 1 p E F S n 1 k p 1 k n X 8 a.s. for p >. Smlarly, 8 remans vald for E 1 k n S k p.usng7agan, from 8 we get E F S k p p+1 1 k n E F X p + p p E F S n 1 k p 1 k n a.s.for p >. But by the condtonal verson of the Hölder nequalty c.f. Loéve [11], we have p p E F S n 1 k p 1 k n E F S k p 1 k n X p /p E F n 1 p p X p/ n 1 p/ p 1 p E F S k p + p +1 p p 1 E F X. 1 k n Here we have used the elementary nequalty a α b β αa + βb for non-negatve numbers a, b,α,β wth α + β = 1. Puttng ths nequalty nto 9, we get then /p X 9
458 D.-M. Yuan et al. E F S k p C p 1 k n E F X p + E F X p/ a.s.. 10 Here and n the sequel, the constant C p dependng only p may be dfferent at each appearance. Let X + = 0, X, X = 0, X. Set ζ = X + E F X +,δ = X E F X, 1 n. Wehave E F X p/ X + X p/ = E F p/ C p E F X + + X p/ C p EF X + p/ + E F X C p EF ζ + E F p/ X + + E F δ + E F p/ X C p EF p/ p/ p/ ζ + E F δ + E F X a.s.. 11 To complete the proof of, we frst consder the specal case < p, thus 1 < p/. It s noted that X + and X are nondecreasng and nonncreasng functons of X, respectvely, so that {ζ, 1} and {δ, 1} are both condtonally centered F-NA random varables by P. Hence p/ E F ζ C p E F ζ p/ { C p E F X + E F X + } p/ { } p/ C p E F X p + E F X p/ C p E F X p + E F X a.s. 1
Condtonal lmt theorems for condtonally negatvely assocated random varables 459 by 1, and analogously for the term correspondng to E F n δ p/.from10, 11 and 1, relaton follows n the specal case < p. We suppose that relaton struefor k < p k+1 where k 1. For k+1 < p k+, that s k < p/ k+1, by the nducton hypothess, p/ E F ζ p/4 C p E F ζ p/ + E F ζ p/ p/4 C p E F X p + E F X + E F X 4 a.s.. 13 Notng that p > 4, we conclude that E F X 4 = whch yelds [ p 4/p E F X X p /p ] { p 4/p E F X E F X p } /p E F X p 4/p /p E F X p a.s., p/4 pp 4/4p p/p E F X 4 E F X E F X p p/ C p E F X + E F X p a.s.. 14 By 13 and 14, E F p/ ζ C p p/ E F X p + E F X a.s.. 15 Smlarly, 15 remans vald for E F n δ p/.from10 and 15, relaton s proved for the general case of k+1 < p k+, thus the proof s complete.
460 D.-M. Yuan et al. 3 Condtonal mean convergence theorems Chandra and Goswam [4] ntroduced a specal type of unform ntegrablty called resdual Cesàro alpha-ntegrablty. More recent dscusson on ths topc can be found n Yuan et al. [0]. Yuan and Tao [19] extended ths concept to resdual h-ntegrablty, whch s weaker than the exstng unform ntegrablty. Now we further weaken unform ntegrablty. Defnton For α>0 fxed, a sequence of random varables {X n, n 1} s sad to be condtonally resdually Cesàro alpha-ntegrable gven F F-RCIα, n short f 1 sup n 1 n E F 1 X < a.s. and lm n n E F [ X α I X > α ] =0a.s.. Clearly, f {X n, n 1} have dentcal condtonal dstrbutons that s, E F I X x = E F I X j x a.s. for < x < wth E F X 1 < a.s., then {X n } s F-RCIα for any α>0. Let p > 1 and let hx be a strctly postve functon defned on 1, +. In ths secton, we dscuss condtonal mean convergence theorems of the form of n hp 1 n S ES for F-NA random varables {X n, n 1}, provded that { X n p, n 1} s F-RCIα for an approprate α. Our frst result s dealng wth the case 1 < p <. Theorem Let 1 < p <, and let {X n, n 1} be a sequence of F-NA random varables. If { X n p, n 1} s F-RCIα for some α 0, 1/p, then condtonally on F n 1/p S E F S 0 n L p. Proof Let 1 n Y n = n α I X n < n α + X n I X n n α + n α I X n > n α, n 1, and for each n 1, defne Z n = X n Y n, S 1 n = Y and S n = Z. It s easy to see that Y n =mn { X n, n α }, Z n = X n n α I X n > n α, and 16 Z n p X n p n α I X n p > n α 17 for all p > 1. Moreover, {Y n E F Y n, n 1} and {Z n E F Z n, n 1} are both sequences of condtonally centered F-NA random varables n vew of P.
Condtonal lmt theorems for condtonally negatvely assocated random varables 461 For our purpose, t suffces to prove n /p E F S 1 1 n E F S 1 0 a.s. 18 and n 1 E F S 1 n E F S p 0 a.s.. 19 Usng relatons 1, 17 and the second condton of the F-RCIα property 16 of the sequence { X n p, n 1}, we obtan n 1 E F S 1 n E F S p n 1 E F Z E F p Z n 1 E F Z p n 1 E F [ X p α I X p > α ] 0 a.s.. ths proves 19. Here and n the sequel, represents the Vnogradov symbol O. For 18, usng relaton 1 agan,wehave n /p E F S 1 1 n E F S 1 n /p n /p E F Y E F Y E F Y n /p n pα E F X p n pα 1/p sup n 1 n 1 E F X p a.s.. Usng the frst condton of the F-RCIα property 16 of the sequence { X n p, n 1}, the last expresson above clearly goes to 0 as n, because p < and α<1/p, thus completng the proof. Next we consder the case p =. Theorem 3 Let {X n, n 1} be a sequence of F-NA random varables. If { X n p, n 1} s F-RCIαforsomeα 0,, then, for any δ>0, condtonally on F n 1/+δ 1 n S ES 0 n L.
46 D.-M. Yuan et al. Proof The proof s smlar to that of Theorem. It suffces to prove that condtonally on F n 1/+δ S 1 E F S 1 0 n L 1 n and n 1/+δ S 1 n E F S 0 n L. These convergences can be establshed by usng relaton 1, and the detals are omtted. For the case p >, we have Theorem 4 Let p >, and let {X n, n 1} be a sequence of F-NA random varables. If { X n p, n 1} s F-RCIαforsomeα 0, 1 /p, then condtonally on F n 1/q S E F S 0 n L p, 1 n where q := p/p 1 s the dual number of p. Proof Proceedng as n the proof of Theorem, we need to show that condtonally on F n 1/q S 1 E F S 1 0 nl p 0 1 n and n 1/q S 1 n E F S 0 nl p. 1 Usng relaton, the Hölder nequalty, relaton 17 and the second condton of the F-RCIα property 16 of the sequence { X n p, n 1}, we obtan n p/q E F S 1 n E F S p p/ Z E F Z n p/q E F Z E F p Z + n p/q E F n p/q Z E F p Z + n p/ E F E F Z E F p Z n p/ n 1 E F Z p E F [ X p α I X p > α ] 0 a.s.,
Condtonal lmt theorems for condtonally negatvely assocated random varables 463 ths proves 1. For 0, usng relaton agan,wehave n p/q E F n p/q n p/q n p 1 S 1 1 n E F S 1 p E F Y E F p Y + n p/q E F E F Y p + n p 1 E F Y p n p 1+pα E F X p n p+pα+ sup n 1 n 1 E F Y p E F X p a.s.. p Y E F Y Usng the frst condton of the F-RCIα property 16 of the sequence { X n p, n 1}, the last expresson above clearly goes to 0 a.s. as n, because p > and α<1 /p, thus completng the proof. 4 Condtonally complete convergence A sequence of random varables {X n, n 1} s sad to converge completely to a constant a f for every ε>0, P X a >ε<. Ths concept was defned by Hsu and Robbns [6]. Note that complete convergence mples almost sure convergence n vew of the Borel Cantell lemma. Now we extend ths concept. Defnton 3 A sequence of random varables {X n, n 1} s sad to be condtonally converge completely gven F to a constant a f P X a >ε F < for every ε>0, and we wrte X n a condtonally completely gven F. Chandra and Goswam [4] ntroduced a specal type of unform ntegrablty called strongly resdual Cesàro alpha-ntegrablty, and some of ts applcatons can be found n Yuan et al.[0] and Yuan and An [1]. Now we further extend ths concept. Defnton 4 For α>0 fxed, a sequence {X n, n 1} of random varables s sad to be condtonally strongly resdually Cesàro alpha-ntegrable gven F F-SRCIα, n
464 D.-M. Yuan et al. short f 1 sup n 1 n E F X < a.s. and n=1 1 n EF [ X n n α I X n > n α ] < a.s.. That the condton of F-SRCIα s a strong verson of the condton of F-RCIα s an easy consequence of the Kronecker lemma. In ths secton, we wll show that each of the theorems n the prevous secton has a correspondng strong analogue n the sense of condtonal complete convergence. We gve a lemma pror to statng our condtonal complete convergence results. Lemma 1 For sequences {a n, n 1} and {b n, n 1} of non-negatve real numbers, f sup n 1 n 1 a < and b n <, n=1 then m a b sup m 1 a m 1 b for every n 1. Proof Let {a, 1 n} and {b, 1 n} be, respectvely, the rearrangements of {a, 1 n} and {b, 1 n} satsfyng a 1 a a n and b 1 b b n. Then n a b n a b. So wthout loss of generalty, one can assume that {a, 1 n} and {b, 1 n} are nonncreasng. By applyng Remark 3 n Landers and Rogge [8], and by usng the monotoncty of {a n } and {b }, one can complete the rest of the proof. Theorem 5 Let 1 < p <, and let {X n, n 1} be a sequence of F-NA random varables. If { X n p, n 1} s F-RCIαforsomeα 0, 1/p, then n 1/p S E F S 0 condtonally completely gven F. 1 n Proof For each n 1, let m = m n be the nteger such that m 1 < n < m. Observe that n 1/p S E F S n 1/p S E F S 1 n 1 m m 1 1/p S E F S 1 m = 1/p m/p 1 m S E F S.
Condtonal lmt theorems for condtonally negatvely assocated random varables 465 Hence t suffces to prove m/p 1 m S E F S 0 condtonally completely gven F. Let Y n, Z n, S n 1 and S n be defned as n the proof of Theorem. We frst prove that m/p S 1 m E F S 0 completely condtonally gven F, namely m/p Z k E F Z k 0 condtonally completely gven F. 3 1 m k=1 Usng relatons 1, 17 and the second condton of the F-SRCIα property of the sequence { X n p },wehave E F m/p Z 1 m k E F Z k = m m k=1 E F Z p E F Z p m: m 1 E F Z p m p 1 E F [ X p α I X p > α ] < a.s., whch yelds 3. Next we show that m/p S 1 1 m E F S 1 0 condtonally completely gven F, namely m/p Y 1 m k E F Y k k=1 p 0 condtonally completely gven F. 4
466 D.-M. Yuan et al. By relaton 1 E F m/p 1 m Y k E F Y k k=1 m m/p m m/p E F Y pα E F X p. In vew of the frst condton of the F-SRCIα property of the sequence { X n p } and Lemma 1, we conclude that = E F m/p 1 m m m/p pα m: m /p+ pα Y k E F Y k k=1 pα 1 p1/p α /p m The last seres above converges because p < and α<1/pand therefore 4 holds. Ths completes the proof. Theorem 6 Let {X n, n 1} be a sequence of F-NA random varables. If { X n, n 1} s F-SRCIαforsomeα 0,, then, for every δ>0, n 1/+δ S E F S 0 condtonally completely gven F. 1 n a.s.. Proof The proof s smlar to that of Theorem 5. It suffces to prove E F m1/+δ Z k E F Z k < 1 m and k=1 E F m1/+δ Y 1 m k E F Y k <. These results can be proved by usng relaton 1, and the detals are omtted. k=1
Condtonal lmt theorems for condtonally negatvely assocated random varables 467 Theorem 7 Let p >, and let {X n, n 1} be a sequence of F-NA random varables. If { X n p, n 1} s F-SRCIα forsomeα 0, 1 /p, then n 1/q S E F S 0 condtonally completely gven F. 1 n Proof Proceedng as n the proof of Theorem 5, we need only to show that m/q 1 m k=1 Z E F Z 0 condtonally completely gven F 5 and m/q 1 m k=1 Y E F Y 0 condtonally completely gven F 6 reman true. By relaton and the Hölder nequalty, E F m/q 1 m m mp/q p Z k E F Z k k=1 E F Z p + mp/q E F mp/q E F m mp/q E F Z p m Z E F Z p m: m p/ E F Z p 1 E F Z p a.s.. mp/ p/ m Z p/
468 D.-M. Yuan et al. Usng relaton 17 and the second condton of the F-SRCIα property of the sequence { X n p }, we conclude that E F m/q Z 1 m k E F Z k k=1 p 1 E F [ X p α I X p > α ] <, whch yelds 5. As for 6, by relaton agan and the Hölder nequalty, E F m/q 1 m m mp/q k=1 Y k E F Y k p E F Y E F p Y + mp/q E F mp/q E F m m mp/q+mp m m m p 1 p Y E F Y E F Y E F p Y E F Y p m m p 1 αp E F X p a.s.. m Y E F Y In vew of the frst condton of the SRCIα property of the sequence { X n p } and Lemma 1, we conclude that E F m/q 1 m p Y k E F Y k k=1 m m p 1 αp m: m p+1+αp αp m p 1 The last seres converges snce α 0, 1 /p mples p + 1 + αp < 1 and so 6 holds. a.s.. p
Condtonal lmt theorems for condtonally negatvely assocated random varables 469 5 Condtonal central lmt theorem The followng lemma s a condtonal verson of Theorem 1 of Newman and Wrght [13], whch s of ndependent nterest. Lemma Suppose that {X, 1 n} s a set of F-NA random varables wth E F X < for each 1 n. Then for any real number t, n EF exp t X j E F exptx j 1 t Cov F X j, X k. j=1 j=1 1 j =k n Proof The result s true for n = 1 trvally and for n = from the observaton c.f. Prakasa Rao [15] Cov F exptx 1, exptx t Cov F X 1, X. So, the result follows by nducton, and by usng the fact that f X,Y and Z are F-NA then so are txare ty + Z by P1 as they are nondecreasng or nonncreasng functons of F-NA random varables accordng to t > 0ort < 0. The next lemma s a precse restatement of Theorem 8 n Prakasa Rao [15]. Lemma 3 Suppose that the random varable X satsfy E F X = 0,τF = EF X <. Then for every F-measurable random varable T, E F [ ] n exptx/τ F n exp T / a.s. as n. Proof We frst fx a real number t.let{x, X n, n 1} be a sequence of F-ndependent random varables wth dentcal condtonal dstrbutons and let S n = n X.By Theorem 8 n Prakasa Rao [15], E F [ expts n /τ F n ] exp t / a.s. as n. 7 On the other hand, we can prove that E F e ty 1+Y Z = E F e ty 1 Z E F e ty Z, provded that Y 1 and Y are F-ndependent and Z s F-measurable by takng frst Z to be a smple random varable and then by usng the usual approxmaton. Further E F [ ] n expts n /τ F n = E F [ ] exptx j /τ F n j=1 = E F [ ] n exptx/τ F n 8
470 D.-M. Yuan et al. by nducton on n. So, t follows that E F [ ] n exptx/τ F n exp t / a.s. as n by combnng 7 and 8. Next, the desred result can be completed by takng T = k j=1 t j I A j, and by usng the usual approxmaton. Our condtonal central lmt theorem stated n terms of condtonal characterstc functons reads as follows. Theorem 8 Let {X n, n 1} be a sequence of F-statonary and F-NA random varables wth E F X 1 = 0, E F X1 < a.s. Then σf := EF X1 + Cov F X 1, X 0 a.s., 9 = n 1 E F S n σ F a.s., 30 and the seres converges almost surely. If σ F > 0 almost surely, then [ E F e ts ] n/σ F n e t / a.s. as n. 31 Proof Relaton 9 s a drect consequence of 30. So, we only need to prove relatons 30 and 31. By the condtonal statonarty, m E F Sm = mef X1 + m j + 1Cov F X 1, X j. j= Note that Cov F X 1, X j 0 for all j. We get whch mples 0 lm m 1 j 1 Cov F X 1, X j E F X m 1 m, j= 0 lm m lm [m/] j= [m/] m E F X 1, j= Cov F X 1, X j 1 j 1 Cov F X 1, X j m
Condtonal lmt theorems for condtonally negatvely assocated random varables 471 that s, the seres m j= Cov F X 1, X j converges almost surely. Further m m j 1 Cov F X 1, X j 0 j= a.s. as m by the Kronecker lemma. Hence 1 m EF S m σ F = j=m+1 + m j= Cov F X 1, X j m j 1 Cov F X 1, X j 0 a.s. asm. Therefore 30 holds. Next we prove 31. Let ψ F n r = EF [ exprs n /σ F n ]. Relaton 31 turns nto ψ F n r exp r / a.s. as n. 3 For any fxed l = 1,,...,defne m as the greatest nteger less than or equals to n/l. Note that ψn F r ψf ml r r /σf [E F S n / n S ml / ml ] 1/ and that n vew of relaton 1, E F Sn S ml = E F 1 1 n S n S ml 1 S ml n ml ml n E F 1 n S n S ml + E F 1 1 S ml ml n n ml n = n ml + 1 n = 1 E F X 1 + 1 ml 1 n mle F X 1 ml E F X1 n ml E F X1 0 a.s. asn, n so ψ F n r ψf ml r 0 a.s. as n. 33
47 D.-M. Yuan et al. Next we defne Y l j = S jl S j 1l /σ F l for j = 1,...,m wth S0 = 0, so that S ml /σ F ml = Y l 1 + +Ym l / m. Note that the famly {Y l j, 1 j m} s F-NA by P1 and P. It follows from the condtonal statonarty and from Lemma wth X j = Y l j and t = r/ m, that ψml F r ψl F m r/ m = r m EF exp m 1 = r EF 1 j =k m j=1 Y l j m j=1 r /mcov F Y l j, Y l k Y l 1 + +Y l m m 1 m E [exp F m E F Y l j j=1 = r [ σf E F S ml / m E F S l / ] l j] r Y l m r σf σf σ l,f a.s. as m, 34 where σ l,f = EF S l / l. Settng Y 1 = S l, and applyng Lemma 3 wth τ F = l 1 σ l,f = EF S l,wehave ψl F m r/ m = = [ E F exp E F [ exp ] m ry 1 / σ F ml exp τf r / lσf τf r Y 1 / ] m τ F m lσf Therefore for any fxed l, we have by combnng 33, 34 and 35, lm sup ψ F n r exp r / n = r σ F a.s. as m. 35 σ F σ l,f + exp τ F r /lσ F exp r /. Snce σl,f = l 1 E F Sl σ F and τ F /lσ F 1asl by relaton 30, ths yelds 3 as desred and completes the proof of Theorem 8. Acknowledgments The authors would lke to sncerely thank the referees for ther valuable comments and suggestons on a prevous draft, whch resulted n the present verson of ths paper. The frst and second authors are supported by the Natonal Natural Scence Foundaton of Chna 1087117 and by the Natural Scence Foundaton Project of CQ CSTC Of Chna 009BB370.
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