CHOVER-TYPE LAWS OF THE ITERATED LOGARITHM FOR WEIGHTED SUMS OF ρ -MIXING SEQUENCES

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1 CHOVER-TYPE LAWS OF THE ITERATED LOGARITHM FOR WEIGHTED SUMS OF ρ -MIXING SEQUENCES GUANG-HUI CAI Receved 24 September 2004; Revsed 3 My 2005; Accepted 3 My 2005 To derve Bum-Ktz-type result, we estblsh Chover-type lw of the terted logrthm for the weghted sums of ρ -mxng nd dentclly dstrbuted rndom vrbles wth dstrbuton n the domn of stble lw. Our result obtned not only generlzes the mn results of Peng nd Q 2003 nd Q nd Cheng 996 to ρ -mxng sequences of rndom vrbles, but lso mproves them. Copyrght 2006 Gung-hu C. Ths s n open ccess rtcle dstrbuted under the Cretve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, nd reproducton n ny medum, provded the orgnl work s properly cted.. Introducton Let {X, } be ndependent nd dentclly dstrbuted..d. wth symmetrc stble dstrbutons, whch belong to the domn of norml ttrcton nd nongenerton. So, ther chrcterstc functons re of the forms: Chover [4] hs obtned tht Eexp tx = exp t α, t R,.. n X / log logn = e /α.s..2 n /α We cll ths Chover-type LIL lws of the terted logrthm. Ths type LIL hs been estblshed by Vsudev nd Dvn [3], Znchenko [4] for delyed sums, by Chen nd Hung [3] for geometrc weghted sums, nd by Chen [2] for weghted sums. Q nd Cheng [] extended the Chover-type lw of the terted logrthm for the prtl sums to the cse where the underlyng dstrbuton s n the domn of ttrcton of nonsymmetrc stble dstrbuton see below for detls. Let L α denote stble dstrbuton wth exponent α 0,2. Recll tht the dstrbuton of X s sd to be n the domn of ttrcton of L α f there exst some constnts A n R Hndw Publshng Corporton Journl of Appled Mthemtcs nd Stochstc Anlyss Volume 2006, Artcle ID 65023, Pges 7 DOI 0.55/JAMSA/2006/65023

2 2 Chover-type LIL for weghted sums of mxng sequences nd > 0suchtht S n A n d L α..3 Under.3, Q nd Cheng [] nd Peng nd Q [0] showed tht n X A n / log logn = e /α.s..4 It s well known tht.3 holds f nd only f Fx = C xlx x α, F x = C 2xlx x α, forx>0,.5 where, for x>0, C x 0, lm x C x = C, =,2, C + C 2 > 0, nd lx 0sslowly vryng n the sense of Krmt functon, tht s, ltx lm =, for x>0..6 t lt By Ln et l. [6, pge 76, Exercse 2], we hve = nln /α. For nonempty sets S,T,wedefne S = σx k, k S. And we defne the ml correlton coeffcent ρn = supcorr f,g where the supremum s tken over ll S,T wth dsts,t n nd for ll f L 2 S, g L 2 T, nd dsts,t = nf x S, y T x y. A sequence of rndom vrbles {X n, n } on probblty spce {Ω,,P} s clled ρ -mxng f lm ρ n = 0..7 As for ρ -mxng sequences of rndom vrbles, one cn refer to Bryc nd Smolensk [], who estblshed bounds for the moments of prtl sums for sequence of rndom vrbles stsfyng lm ρ n <..8 Pelgrd [7] estblshed CLT. Pelgrd [8] estblshed n nvrnce prncple. Pelgrd nd Gut [9] estblshed Rosenthl-type ml nequltes nd Bum-Ktz-type results. Utev nd Pelgrd [2] estblshed n nvrnce prncple of nonsttonry sequences. To derve Bum-Ktz-type result, the mn purpose of ths pper s to estblsh Chover-type lw of the terted logrthm for the weghted sums of ρ -mxng nd dentclly dstrbuted rndom vrbles wth dstrbuton n the domn of stble lw. Our result not only generlzes the mn results of Peng nd Q [0] nd Q nd Cheng [] to ρ -mxng sequences of rndom vrbles, but lso mproves them. Throughout ths pper, let h B[0,] denote tht the functon h s bounded on [0,]. C wll represent postve constnt though ts vlue my chnge from one ppernce to the next, nd = Ob n wllmen Cb n.

3 2. The mn results In order to prove our results, we need the followng lemm nd defnton. Gung-hu C 3 Lemm 2. Utev nd Pelgrd [2]. Let {X, } be ρ -mxng sequence of rndom vrbles, EX = 0, E X p < for some p 2 nd for every. there exsts C =Cp, such tht k p E X C E X p n + k n EX 2 p/2. 2. Defnton 2.2 Ln nd Lu [5]. Afuncton f x > 0x>0 s sd to be qusmonotone nondecresng, f x sup 0 t x f t <. 2.2 f x Here re our mn results. Theorem 2.3. Let {X,X, } be ρ -mxng sequence of dentclly dstrbuted rndom vrbles. Let h be bounded functon on [0,], contnuoustx 0 0,. LetS n = n h/nx, EX = 0, when α>. Let f x > 0 be qusmonotone nondecresng nd /x f xdx <. under condton.3, for ny ε>0, n P S >ε nfnln /α <. 2.3 Proof of Theorem 2.3. For ny, defne X n =X I X, S n = h/nx n /nx n, where = nfnln /α.fornyε>0, P S >ε P X > + P S n >ε n X n. 2.4 Frst we show tht n X n 0, s n. 2.5

4 4 Chover-type LIL for weghted sums of mxng sequences In fct, when 0 <α, h B[0,]. For ny postve ntegers n,n, X n n n E h X n Cn x dfx n n x Cn N + Cn x dfx =: CA + B. N < x 2.6 Snce f x > 0 s qusmonotone nondecresng nd by.5, we hve, for n N, N lrge enough, B = n C n k=n+ k=n+ C f N + C k < x k x dfx n k=n+ k P k < X k kp k < X k CNP X N + C P X k k=n It s obvous tht for ech gven N, kfk C f N + C N N dx kfk < ε 4. k=n 2.7 A C /α 0, s n. 2.8 f n So, for 0 <α, we hve 2.5. When <α<2, usng EX = 0, h B[0,], nd.5, when n,wehve n X n = X I X > n I X X > n n n Cn E X I X > = Cn = n C α n = C f n < ε 2. P X x dx = Cn Cln x α dx 2.9 So, for <α<2, we lso hve 2.5. Hence 2.5holdsfor0<α<2. By 2.4nd2.5, we hvetht P S >ε P X > + P S n > ε 2, 2.0 =

5 Gung-hu C 5 for n lrge enough. Hence we need only to prove II =: I =: n n = P X > <, n P S n > ε 2 <. 2. From.5, t s esly seen tht I = P C X > nfn C dx <. 2.2 xfx By Lemm 2. nd the fct tht h B[0,], t follows tht II C n E S n 2 2 C n n n 2 n n C E X 2 I X = C x 2 dfx x 2 n = C 2 n k= 2 n k= X n 2 x 2 dfx C 2 k P k < X k k < x k C k=kp k < X k C whch completes the proof of Theorem 2.3. dx xfx <, Corollry 2.4. Under the condtons of Theorem 2.3, n=k 2 n 2.3 S n / loglogn e /α.s. 2.4 Proof of Corollry 2.4. Notce tht for ny postve nteger n, there exsts n nonnegtve nteger k,suchtht2 k n<2 k+. And there exsts t [0,, such tht n = 2 k+t.by2.3, we hve 2 k+ k=0 n=2 k 2 k+ P S >ε 2 k+ f 2 k+t l 2 k+t /α 2 k+t <. 2.5 P S >ε 2 k+ f 2 k+t l 2 k+t /α < k+t k=0

6 6 Chover-type LIL for weghted sums of mxng sequences So 2 k+t S 2 k+ f 2 k+t l 2 k+t /α 0.s. 2.7 S n 2 k+t S 2 k+ /α nfnln 2 k+ f 2 k+t l f 2 k+t l 2 k+t /α 2 k+t /α /α nfn 2 /α 2 k+t S 2 k+ f 2 k+t /α 0.s. 2.8 S n /α = 0.s. 2.9 nfnln Gven ε>0, let f x = log +ε x. It s obvous tht /x f xdx <.By2.9, we hve S n nlnlog +ε n /α = 0.s Therefore S n / loglogn e +ε/α.s. 2.2 Bn S n / loglogn e /α.s., 2.22 Bn whch completes the proof of 2.4. Remrk 2.5. Corollry 2.4 generlzes the estmte S n / loglogn e /α.s obtned n Peng nd Q [0, Theorem 2.] to ρ -mxng sequences of rndom vrbles. Corollry 2.6. Under the condtons of Corollry 2.4,lettng hx,yelds n X / loglogn e /α.s Remrk 2.7. Corollry 2.6 generlzes n Q nd Cheng [, Theorem.] to ρ -mxng sequences of rndom vrbles.

7 Gung-hu C 7 References [] W. Bryc nd W. Smoleńsk, Moment condtons for lmost sure convergence of wekly correlted rndom vrbles, Proceedngs of the Amercn Mthemtcl Socety 9 993, no. 2, [2] P. Y. Chen, Lmtng behvor of weghted sums wth stble dstrbutons, Sttstcs & Probblty Letters , no. 4, [3] P. Y. Chen nd L. H. Hung, The Chover lw of the terted logrthm for rndom geometrc seres of stble dstrbuton, ActMthemtcSnc , no. 6, Chnese. [4] J. Chover, A lw of the terted logrthm for stble summnds, Proceedngs of the Amercn Mthemtcl Socety 7 966, no. 2, [5] Z.Y.LnndC.R.Lu,Lmt Theory for Mxng Dependent Rndom Vrbles, Mthemtcs nd Its Applctons, vol. 378, Kluwer Acdemc, Dordrecht; Scence Press, New York, 996. [6] Z. Y. Ln, C. R. Lu, ndz. G. Su, Foundton of Probblty Lmt Theory, Hgher Educton Press, Beng, 999. [7] M. Pelgrd, On the symptotc normlty of sequences of wek dependent rndom vrbles,journl of Theoretcl Probblty 9 996, no. 3, [8], Mxmum of prtl sums nd n nvrnce prncple for clss of wek dependent rndom vrbles, Proceedngs of the Amercn Mthemtcl Socety , no. 4, [9] M. Pelgrd nd A. Gut, Almost-sure results for clss of dependent rndom vrbles, Journlof Theoretcl Probblty 2 999, no., [0] L. Peng nd Y. C. Q, Chover-type lws of the terted logrthm for weghted sums, Sttstcs& Probblty Letters , no. 4, [] Y. C. Q nd P. Cheng, A lw of the terted logrthm for prtl sums n the feld of ttrcton of stblelw, Chnese Annls of Mthemtcs. Seres A. Shuxue Nnkn. A J 7 996, no. 2, Chnese. [2] S. Utev nd M. Pelgrd, Mxml nequltes nd n nvrnce prncple for clss of wekly dependent rndom vrbles, Journl of Theoretcl Probblty , no., 0 5. [3] K. Vsudev nd G. Dvn, LIL for delyed sums under on-dentclly dstrbuted setup, Teory Veroytnosteĭ ee Prmeneny , no. 3, Russn, trnslton n Theory Probb. Appl , no. 3, [4] N. M. Znchenko, A modfed lw of terted logrthm for stble rndom vrbles, Theory of Probblty nd Mthemtcl Sttstcs , Gung-hu C: Deprtment of Mthemtcs nd Sttstcs, Zheng Gongshng Unversty, Hngzhou 30035, Chn E-ml ddress: cghzu@63.com