Statistics and Probability Letters

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1 Sttstcs nd Probblty Letters 79 (2009) Contents lsts vlble t ScenceDrect Sttstcs nd Probblty Letters journl homepge: Lmtng behvour of movng verge processes under ϕ-mxng ssumpton Pngyn Chen, Ten-Chung Hu b,, Andre Volodn c Deprtment of Mthemtcs, Jnn Unversty, Gungzhou, , PR Chn b Deprtment of Mthemtcs, Ntonl Tsng Hu Unversty, Hsnchu 300, Twn, ROC c Deprtment of Mthemtcs nd Sttstcs, Unversty of Regn, Regn, Ssktchewn S4S 0A2, Cnd r t c l e n f o b s t r c t Artcle hstory: Receved 20 October 2006 Receved n revsed form 11 June 2008 Accepted 23 July 2008 Avlble onlne 5 August 2008 MSC: 60F15 Let, < < be doubly nfnte sequence of dentclly dstrbuted ϕ-mxng rndom vrbles,, < < be n bsolutely summble sequence of rel numbers. In ths pper we prove the complete convergence nd Mrcnkewcz Zygmund strong lw of lrge numbers for the prtl sums of movng verge processes X n = +n, n 1 bsed on the sequence, < < of ϕ-mxng rndom vrbles, mprovng the result of [Zhng, L., Complete convergence of movng verge processes under dependence ssumptons. Sttst. Probb. Lett. 30, ] Elsever B.V. All rghts reserved. 1. Introducton nd formulton of the mn results Let, < < + be doubly nfnte sequence of dentclly dstrbuted rndom vrbles nd, < < + be n bsolutely summble sequence of rel numbers. Let X n = +n, n 1 be the movng verge process bsed on the sequence, < < +. As usul, we denote S n = n k=1 X k, n 1, the sequence of prtl sums. Under the ssumpton tht, < < + s sequence of ndependent dentclly dstrbuted rndom vrbles, mny lmtng results hve been obtned for the movng verge process X n, n 1. For exmple, Ibrgmov (1962) estblshed the centrl lmt theorem, Burton nd Dehlng (1990) obtned lrge devton prncple, nd L et l. (1992) obtned the complete convergence result for X n, n 1. Certnly, even f, < < + s the sequence of ndependent dentclly dstrbuted rndom vrbles, the movng verge rndom vrbles X n, n 1 re dependent. Ths knd of dependence s clled wek dependence. The prtl sums of wekly dependent rndom vrbles X n, n 1 hve smlr lmtng behvour propertes n comprson wth the lmtng propertes of ndependent dentclly dstrbuted rndom vrbles. For exmple, we could present some of the prevous results connected wth complete convergence. The followng ws proved n Hsu nd Robbns (1947). Theorem A. Suppose X n, n 1 s sequence of ndependent dentclly dstrbuted rndom vrbles. If EX 1 = 0, E X 1 2 <, then P S n εn < for ll ε > 0. The bove result ws extended by L et l. (1992) for movng verge processes. Correspondng uthor. Tel.: ; fx: E-ml ddress: tchu@mth.nthu.edu.tw (T.-C. Hu) /$ see front mtter 2008 Elsever B.V. All rghts reserved. do: /j.spl

2 106 P. Chen et l. / Sttstcs nd Probblty Letters 79 (2009) Theorem B. Suppose X n, n 1 s the movng verge process bsed on sequence, < < of ndependent dentclly dstrbuted rndom vrbles wth E 1 = 0, E 1 2 <. Then P S n εn < for ll ε > 0. Very few results for movng verge process bsed on dependent sequence re known. In ths pper, we provde two results on the lmtng behvour of movng verge process bsed on ϕ-mxng sequence. Let, < < be sequence of rndom vrbles defned on probblty spce (Ω, F, P) nd denote σ - lgebrs F m n = σ (, n m), n m +. Recll tht sequence of rndom vrbles, < < s clled ϕ-mxng f the mxng coeffcent ϕ(m) = sup sup PB A PB, A F k, PA 0, B F k+m 0 k 1 s m. Recll tht functon h s sd to be slowly vryng t nfnty f t s rel vlued, postve nd mesurble on [0, ), nd f for ech λ > 0 lm x h(λx) h(x) = 1. We refer to Senet (1976) for other equvlent defntons nd for detled nd comprehensve study of propertes of slowly vryng functons. In the followng, we frequently use the followng propertes of slowly vryng functons (cf. Senet (1976)). If h s functon slowly vryng t nfnty, then for ny 0 b nd s 1 b x s h(x) dx x s+1 h(x) b, where C does not depend on nd b, nd for ny λ > 0 mx h(x) (λ)h(λ). x λ Of course, these two nequltes tke plce only f the rght hnd sdes mke sense. The followng result of prtl sums of ϕ-mxng rndom vrbles ws proved n Sho (1988), Remrks 3.2 nd 3.3. Theorem C. Let h be functon slowly vryng t nfnty, 1 p < 2, r 1, nd X, 1 be sequence of dentclly dstrbuted ϕ-mxng rndom vrbles wth EX 1 = 0 nd E X 1 rp h( X 1 p ) <. () If r > 1 then n r 2 h(n)p mx S k εn 1/p <, for ll ε > 0. nd n r 2 h(n)p sup k n Sk /k 1/p ε <, for ll ε > 0. () If r = 1 nd ϕ1/2 (2 m ) <, then h(n) n P mx S k εn 1/p <, for ll ε > 0. For movng verge processes, Zhng (1996) obtned the followng result. Theorem D. Let h be functon slowly vryng t nfnty, 1 p < 2, nd r 1. Suppose tht X n, n 1 s movng verge process bsed on sequence, < < of dentclly dstrbuted ϕ-mxng rndom vrbles wth ϕ1/2 (m) <. If E 1 = 0 nd E 1 rp h( 1 p ) <, then n r 2 h(n)p S n εn 1/p <, for ll ε > 0. Keepng n mnd the bove mentoned nlogy between the usul lmtng behvour of rndom vrbles nd lmtng behvour of the movng verge process (cf. Theorems A nd B), we note tht substntl gp between Theorems C nd D s dstnct. Frstly, when r > 1, Theorem C provdes the result wthout ny mxng rte, even when r = 1 Theorem C requres weker condton on mxng rte thn Theorem D. Secondly, Theorem D does not dscuss the complete convergence for the cse of the mxmums nd supremums of the prtl sums s t s done n Theorem C. Note tht by the method of Zhng (1996) t s mpossble to elmnte these dfferences. The mn gol of the present nvestgton s to obtn the results smlr to Theorem C, but for the movng verge processes nd usng dfferent methods from those n Zhng (1996). Now we stte the mn results. Theorems 1 nd 2 mprove Theorem D nd extend Theorem C on the cse of movng verge processes. The proofs wll be detled n the next secton.

3 P. Chen et l. / Sttstcs nd Probblty Letters 79 (2009) Theorem 1. Let h be functon slowly vryng t nfnty, 1 p < 2, nd r > 1. Suppose tht X n, n 1 s movng verge process bsed on sequence, < < of dentclly dstrbuted ϕ-mxng rndom vrbles. If E 1 = 0 nd E 1 rp h( 1 p ) <, then () nr 2 h(n)p mx S k εn 1/p <, for ll ε > 0. nd () nr 2 h(n)p sup Sk k n /k 1/p ε <, for ll ε > 0. The second theorem trets the cse r = 1. Theorem 2. Let h be functon slowly vryng t nfnty nd 1 p < 2. Assume tht θ <, where θ belong to (0, 1) f p = 1 nd θ = 1 f 1 < p < 2. Suppose tht X n, n 1 s movng verge process bsed on sequence, < < of dentclly dstrbuted ϕ-mxng rndom vrbles wth ϕ1/2 (2 m ) <. If E 1 = 0 nd E 1 p h( 1 p ) <, then h(n) n P mx S k εn 1/p <, for ll ε > 0. In prtculr, the ssumptons E 1 = 0 nd E 1 p < mply the followng Mrcnkewcz Zygmund strong lw of lrge numbers S n /n 1/p 0 lmost surely s n. 2. Few techncl lemms The followng fve lemms wll be useful. The frst two lemms cn be found n Sho (1988) Lemm 3.1 nd Corollry 2.1, hence we omt ther proofs. For the frst two lemms we ssume tht n, n 1 s ϕ-mxng sequence nd S k (n) = k+n =k+1, n 1, k 0. Lemm 1. Let E = 0, E 2 ES 2 (n) k n exp < for ll 1. Then for ll n 1 nd k 0 we hve ϕ 1/2 (2 ) mx E 2. k+1 k+n Lemm 2. Suppose tht there exsts n rry C k,n, k 0, n 1 of postve numbers such tht mx 1 n ES 2() k C k,n for every k 0, n 1. Then for ny q 2, there exsts C = C(q, ϕ( )) such tht for ny k 0, n 1 ( ( )) E mx S k () q C q/2 + k,n E mx q. 1 n k< k+n The next two lemms seem to be known (cf., for exmple the proof of Theorem G n Chen et l. (2006)), but we nclude ther short nd smple proofs for the nterested reder. Here we let h be functon slowly vryng t nfnty. Lemm 3. If r > 1 nd 1 p < 2, then for ny ε > 0 n r 2 h(n)p k 1/p S k ε n r 2 h(n)p sup k n mx S k (ε/2 1/p )n 1/p. Proof. We hve the followng estmtons: 2 m 1 n r 2 h(n)p sup S k /k 1/p > ε = n r 2 h(n)p sup S k /k 1/p > ε k n n=2 m 1 k n 2 m 1 P sup S k /k 1/p > ε 2 m(r 2) h(2 m ) k 2 m 1 n=2 m 1 2 m(r 1) h(2 m )P sup S k /k 1/p > ε k 2 m 1 = C 2 m(r 1) h(2 m )P mx S k /k 1/p > ε 2 l 1 <k 2 l sup l m

4 108 P. Chen et l. / Sttstcs nd Probblty Letters 79 (2009) m(r 1) h(2 m ) P mx S k > ε2 (l 1)/p 1 k 2 l=m l l = C P mx S k > ε2 (l 1)/p 2 m(r 1) h(2 m ) 1 k 2 l=1 l 2 l(r 1) h(2 l )P mx S k > ε2 (l 1)/p 1 k 2 l=1 l 2 l 1 n r 2 h(n)p mx S k > (ε/2 1/p )n 1/p l=1 n=2 l 1 n r 2 h(n)p mx S k > (ε/2 1/p )n 1/p. Lemm 4. Let be rndom vrble wth E rp h( p ) <, where r 1 nd p 1. If q > rp, then n r 1 q/p h(n)e q I n 1/p E rp h( p ). Proof. Snce r q/p < 0, we hve tht n n r 1 q/p h(n)e q I n 1/p = n r 1 q/p h(n) E q Im 1 < p m = E q Im 1 < p m n r 1 q/p h(n) n=m m r q/p h(m)e q Im 1 < p m Em r q/p h(m) q Im 1 < p m E( p ) r q/p h( p ) q Im 1 < p m E rp h( p )Im 1 < p m E rp h( p ). The lst lemm presents techncl fct tht s mportnt n the proofs of Theorems 1 nd 2. Lemm 5. Let h be functon slowly vryng t nfnty nd p 1. Suppose tht X n, n 1 s movng verge process bsed on sequence, < < of men zero dentclly dstrbuted rndom vrbles such tht E 1 p <. For ny ε > 0 denote I =: n r 2 h(n)p mx j I j > n 1/p εn1/p /2 nd J =: n r 2 h(n)p mx εn 1/p /4, where = j I j n 1/p E j I j n 1/p.

5 P. Chen et l. / Sttstcs nd Probblty Letters 79 (2009) If I < nd J <, then n r 2 h(n)p mx S k εn 1/p I + J <. Proof. Note tht n n X k = k=1 k=1 +k = nd snce <, n 1/p E j j I j n 1/p = n 1/p E n 1/p Hence for n lrge enough we hve n 1/p E j I j n 1/p < ε/4. Then j I j > n 1/p E j I j > n 1/p ( ) n 1 1/p E 1 I 1 > n 1/p E(n 1/p ) p 1 1 I 1 > n 1/p E 1 p I 1 > n 1/p 0, s n. n r 2 h(n)p mx S k εn 1/p n r 2 h(n)p mx + C n r 2 h(n)p mx = I + J. (E j = 0) j I j > n 1/p εn1/p /2 εn 1/p /4 3. Proof of mn results Wth ll the prerequstes ccounted before, we could now prove the mn results of the pper. We strt wth Theorem 1. Proof. Accordng to Lemm 3 t s enough to show tht () holds. Accordng to Lemm 5 t s enough to prove tht I < nd J <. For I, by Mrkov nequlty we hve I n r 2 h(n)n 1/p E mx n r 1 1/p h(n)e 1 I 1 > n 1/p j I j > n 1/p = C n r 1 1/p h(n) E 1 Im < 1 p m + 1 m=n m = C E 1 Im < 1 p m + 1 n r 1 1/p h(n) m r 1/p h(m)e 1 Im < 1 p m + 1 E 1 rp h( 1 p ) <.

6 110 P. Chen et l. / Sttstcs nd Probblty Letters 79 (2009) For J, by Mrkov nd Hölder nequltes, Lemms 1 nd 2, we hve tht for ny q 2 J n r 2 h(n)n q/p q E mx ( ( ( )) n r 2 h(n)n q/p E 1 1/q) 1/q q mx ( ) q 1 n r 2 (q/p) q h(n) E mx ( ) q/2 n r 2 (q/p) h(n) n exp 6 ϕ 1/2 (2 ) (E 1 2 I 1 n 1/p ) q/2 + C n r 1 (q/p) h(n)e 1 q I 1 n 1/p =: J 1 + J 2. Note tht ϕ(m) 0 s m, hence ϕ 1/2 (2 ) = o(log n). Furthermore, exp A ϕ 1/2 (2 ) = o(n t ) for ny A > 0 nd t > 0. We consder two seprte cses. If rp < 2, tke q = 2. Note tht n ths cse r (2r/p) < 0. Tke t > 0 smll enough such tht r (2r/p) + t < 0. We hve J 1 = C ( n r 2 (2/p) h(n) n exp 6 n r (2/p) 1 h(n) exp 6 [log n] [log n] ϕ 1/2 (2 ) ϕ 1/2 (2 ) ) n r 2/p+t 1 h(n)e 1 rp 1 2 rp I 1 n 1/p n r (2r/p)+t 1 h(n)e 1 rp <. E 1 2 I 1 n 1/p E 1 2 I 1 n 1/p If rp 2, tke q > 2p(r 1). We hve tht r (q/p) + (q/2) < 1. Next, tke t > 0 smll enough such tht 2 p r (q/p) + (q/2) + t < 1. Note tht n ths cse E 1 2 <. We hve ( ) q/2 J 1 = C n r 2 (q/p) h(n) n exp 6 ϕ 1/2 (2 ) (E 1 2 I 1 n 1/p ) q/2 n r (q/p)+(q/2)+t 2 h(n) <. By Lemm 4 we hve tht J 2 <. Next, we prove Theorem 2. Proof. By Lemm 5 we only need to show tht I < nd J < wth r = 1. For I, by Mrkov nd C r -nequltes (note tht θ 1) I n 1 h(n)n θ/p θ E mx j I j > n 1/p n θ/p h(n)e 1 θ I 1 > n 1/p = C n θ/p h(n) E 1 θ Im < 1 p m + 1 m=n

7 m = C E 1 θ Im < 1 p m + 1 n θ/p h(n) P. Chen et l. / Sttstcs nd Probblty Letters 79 (2009) m 1 θ/p h(m)e 1 θ Im < 1 p m + 1 E 1 p h( 1 p ) <. For J, by Mrkov nd Hölder nequltes, nd Lemm 1 J n 1 h(n)n 2/p 2 E mx ( )) n 1 h(n)n 2/p E ( 1/2 1/2 2 mx ( ) n 1 2/p 2 h(n) E mx ( ) [log n] n 1 2/p h(n) n exp 6 ϕ 1/2 (2 ) E 1 2 I 1 n 1/p n 2/p h(n)e 1 2 I 1 n 1/p <. The lst nequlty holds by Lemm 4. Now we wll show lmost sure convergence. By the frst prt of Theorem 2, E 1 = 0 nd E 1 p < mply n 1 P mx S k εn 1/p <, for ll ε > 0. Hence > n 1 P mx S m > εn 1/p 1 m n 2 k = n 1 P k=1 n=2 k 1 1/2 P k=1 By Borel Cntell lemm, 2 k/p mx 1 m 2 k S m 0 mx S m > εn 1/p 1 m n mx S m > ε2 k/p 1 m 2 k 1 lmost surely whch mples tht S n /n 1/p 0 lmost surely. Acknowledgements. The reserch of P. Chen hs been supported by the Ntonl Nturl Scence Foundton of Chn. The reserch of T.-C. Hu hs been prtlly supported by the Ntonl Scence Councl of R.O.C. The reserch of A. Volodn hs been prtlly supported by the Ntonl Scence nd Engneerng Reserch Councl of Cnd. References Burton, R.M., Dehlng, H., Lrge devtons for some wekly dependent rndom processes. Sttst. Probb. Lett. 9, Chen, P., Hu, T.-C., Volodn, A., A note on the rte of complete convergence for mxmums of prtl sums for movng verge processes n Rndemcher type Bnch spces. Lobchevsk J. Mth. 21, Hsu, P.L., Robbns, H., Complete convergence nd the lw of lrge numbers. Proc. Ntl. Acd. Sc. USA 33, Ibrgmov, I.A., Some lmt theorem for sttonry processes. Theory Probb. Appl. 7, L, D., Ro, M.B., Wng, X.C., Complete convergence of movng verge processes. Sttst. Probb. Lett. 14, Senet, E., Regulrly Vryng Functon. In: Lecture Notes n Mth., vol Sprnger, Berln. Sho, Q.M., A moment nequlty nd ts pplcton. Act Mth. Snc 31, (n Chnese). Zhng, L., Complete convergence of movng verge processes under dependence ssumptons. Sttst. Probb. Lett. 30,