Marcinkiewicz strong laws for linear statistics of ρ -mixing sequences of random variables

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1 Aas da Academa Braslera de Cêcas : Aals of the Brazla Academy of Sceces ISSN Marckewcz strog laws for lear statstcs of ρ -mxg sequeces of radom varables GUANG-HUI CAI Departmet of Mathematcs ad Statstcs, Zheag Gogshag Uversty, Hagzhou P. R. Cha Mauscrpt receved o Aprl 4, 2006; accepted for publcato o August st, 2006; preseted by MANFREDO DO CARMO ABSTRACT Strog laws are establshed for lear statstcs that are weghted sums of a radom sample. We show extesos of the Marckewcz-Zygmud strog laws uder certa momet codtos o both the weghts ad the dstrbuto. These ot oly geeralze the result of Ba ad Cheg 2000, Statst Probab Lett 46: 05 2 to ρ -mxg sequeces of radom varables, but also mprove them. Key words: ρ -mxg, Marckewcz-Zygmud strog laws, weghted sums. INTRODUCTION As Ba ad Cheg 2000 remarked, may useful lear statstcs based o a radom sample are weghted sums of..d. radom varables. Examples clude least-squares estmators, oparametrc regresso fucto estmators ad ackkfe estmates, amog others. I ths respect, studes of strog laws for these weghted sums have demostrated sgfcat progress probablty theory wth applcatos mathematcal statstcs. But a radom sample s ofte depedet. So we wat to kow f the results obtaed for..d. radom varables are stll true for ρ -mxg sequeces of radom varables. Let S, T N be oempty ad defe F S = σx k, k S, ad the maxmal correlato coeffcet ρ = sup corr f, g where the supremum s take over all S, T wth dsts, T ad all f L 2 F S, g L 2 F T ad where dsts, T = f x S,y T x y. A sequece of radom varables {X, } o a probablty space {,F, P} s called ρ -mxg f lm ρ <.. E-mal: cghzu@63.com AMS Classfcato: 60F5; 62G05

2 66 GUANG-HUI CAI As for ρ -mxg sequeces of radom varables, oe ca refer to Bryc ad Smolesk 993, who foud bouds for the momets of partal sums for a sequece of radom varables satsfyg., ad to Pelgrad 996 for CLT, Pelgrad 998 for varace prcples, Pelgrad ad Gut 999 for the Rosethal type maxmal equalty, Utev ad Pelgrad 2003 for varace prcples of ostatoary sequeces. The ma purpose of ths paper s to establsh the Marckewcz- Zygmud strog laws for lear statstcs of ρ -mxg sequeces of radom varables. The results obtaed see Theorem 2. ad Corollary 2. ot oly geeralze the result of Ba ad Cheg 2000 to ρ -mxg sequeces of radom varables, but also mprove them. I Theorem 2.2 of Ba ad Cheg 2000, they beleve the choce of b ca hardly be mproved vew of Cuzck 995, Lemma 2., but ow we mprove the choce of b usg a ew method. 2 THE MARCINKIEWICZ-ZYGMUND STRONG LAWS Throughout ths paper, C wll represet a postve costat though ts value may chage from oe appearace to the ext, ad a = Ob wll mea a b. I order to prove our results, we eed the followg lemma. LEMMA 2.. Utev ad Pelgrad, Let {X, } be a ρ -mxg sequece of radom varables, E X = 0, E X p < for some p 2 ad for every. The there exsts C = Cp, such that k { p/2 } E max X p E X p + EX 2. k THEOREM 2.. Let {X, X, } be a ρ -mxg sequece of detcally dstrbuted radom varables, T = a X,, where the weghts {a,, } are radom varables whch are depedet of {X, } the case of determstc weghts s cluded. Suppose that for some α wth 0 <α<2we have that a α = O almost surely. If <α<2, we assume addtoally that E X = 0. Set b = α log. We assume that for some h, > 0, we have E exph X <. 2.0 The ε >0, P max >εb <. 2. PROOF., defe X = X I X b, T = a X Ea X, the ε >0,

3 LINEAR STATISTICS OF ρ -MIXING SEQUENCES OF RANDOM VARIABLES 67 we have P max >εb P max X > b + P max + P max X > b + P max Ea X >εb >εb max Ea X. 2.2 Frst we show that b max By a α = O ad Hölder equalty, k <α, the Ea X 0, as. 2.3 k a k a k α α α k α k. 2.4 Whe <α<2, usg EX = 0, 2.4, Markov equalty ad 2.0, whe, the b max b = b Ea X E a X I X > b a E X I X > b b E X I X > b = Cb E X I b k < X b k+ k= b b b b k+ P X > b k k= k= b k+ E exph X exphb k k + α logk + k hk/α k= α log 2.5 = C α log 0.

4 68 GUANG-HUI CAI a α = O mples that max a =O α. By ths ad Hölder equalty, k α, the a k = a α a k α k α α = C k α. 2.6 Whe 0 <α, usg 2.6, Markov equalty ad 2.0, whe, the b max b = b Ea X E a X I X b a E X I X b b α E X I X b = Cb α E X I b k < X b k b α b α b α b k P X > b k E exph X b k exphb k k α log k k hk /α 2.7 α log α = Clog 0. From 2.5 ad 2.7, Hece 2.3 s true. From 2.2 ad 2.3, t follows that for large eough P max >εb Hece we eed oly to prove that I =: II =: = P X > b + P max > ε 2 b. P X > b <, = P max > ε b <.

5 LINEAR STATISTICS OF ρ -MIXING SEQUENCES OF RANDOM VARIABLES 69 From the fact that E exph X <, t follows easly that I = = P X > b P X > b E exph X exphb <. h/α 2.9 By Lemma 2., t follows that for q 2 II =2 =2 =: II + II 2. b q E max q { b q E a X q + = = q/2} E a X Let max2,α, + q, usg 2.6, we have II = C = C =2 =2 =2 b q =k a q E X q I X b b q q α E X q I X b b q q α E X q I b k < X b k + q α q/α log q/ b q k P X > b k b q k E exph X exphb k k q α log k q k hk /α <. 2.

6 620 GUANG-HUI CAI By 0 <α<2, 2.6 ad q +,wehave II 2 = C = C q/2 b q a 2 E X 2 I X b q/2 b q 2 q/2 α E X 2 I X b q/2 =2 =2 q/2 E X 2 I b k < X b k log q/ =2 bk 2 P X > b k log q/ =2 b 2 k q/2 E exph X q/2 exphb k log q/ =2 log q/ k α 2 log k 2 q/2 exphk /α logk =2 q/2 = C k α 2 2 log k k hk /α log q/ =2 q/2 k 2 log q/ =2 log q/ <. =2 2.2 Puttg 2. ad 2.2 to 2.0 yelds II <. Now we complete the prove of Theorem 2.. COROLLARY 2.. Uder the codtos of Theorem 2., the lm b = 0 a.s. PROOF. By 2., we have > = 2 =0 P max >εb 2 + P max >ε α log =2 P max >ε2 + α log By Borel-Catell Lemma, we have P max >ε2 + α log 2 +,.o. = 0. 2

7 LINEAR STATISTICS OF ρ -MIXING SEQUENCES OF RANDOM VARIABLES 62 Hece ad usg We have lm max 2 α 2 2 <2 b max α log 2 + max 2 = 0 a.s. 2 + α log 2 + lm = 0 a.s. b +, REMARK 2.. Corollary 2. geeralzes the Theorem 2.2 of Ba ad Cheg 2000 to ρ -mxg sequeces of radom varables ad the restrcto of b Corollary 2. s weaker tha the restrcto of b Theorem 2.2 of Ba ad Cheg ACKNOWLEDGMENTS The author would lke to thak a aoymous referee for hs/her valuable commets. Research supported by Natoal Natural Scece Foudato of Cha. RESUMO Les fortes são estabelecdas para estatístcas leares que são somas poderadas de uma amostra aleatóra. Mostramos extesões das les fortes de Marckewcz-Zygmud sob certas codções tato os pesos quato a dstrbução. Estas últmas ão só geeralzam o resultado de Ba e Cheg 2000, Statst Probab Lett 46: 05-2 para sequêcas aleatóras ρ -mxg como também o melhoram. Palavras-chave: ρ -mxg, Marckewcz-Zygmud les fortes, somas poderadas. REFERENCES BAI ZD AND CHENG PE Marckewcz strog laws for lear statstcs. Statst Probab Lett 46: BRYC W AND SMOLENSKI W Momet codtos for almost sure covergece of weakly correlated radom varables. Proc Amer Math Soc 2: CUZICK J A strog law for weghted sums of..d. radom varables. J Theoret Probab 8: PELIGRAD M O the asymptotc ormalty of sequeces of weak depedet radom varables. J Theoret Probab 9: PELIGRAD M Maxmum of partal sums ad a varace prcple for a class weak depedet radom varables. Proc Amer Math Soc 26: PELIGRAD M AND GUT A Almost sure results for a class of depedet radom varables. J Theoret Probab 2: UTEV S AND PELIGRAD M Maxmal equaltes ad a varace prcple for a class of weakly depedet radom varables. J Theoret Probab 6: 0 5.