Weak and strong laws of large numbers for arrays of rowwise END random variables and their applications

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1 Metika 207) 80: DOI 0.007/s z Weak ad stog laws of lage umbes fo aays of owwise ND adom vaiables ad thei applicatios Aitig She Adei Volodi 2 Received: Mach 206 / Published olie: 22 May 207 Spige-Velag Beli Heidelbeg 207 Abstact I the pape, the Macikiewicz Zygmud type momet iequality fo exteded egatively depedet ND, i shot) adom vaiables is established. Ude some suitable coditios of uifom itegability, the L covegece, weak law of lage umbes ad stog law of lage umbes fo usual omed sums ad weighted sums of aays of owwise ND adom vaiables ae ivestigated by usig the Macikiewicz Zygmud type momet iequality. I additio, some applicatios of the L covegece, weak ad stog laws of lage umbes to opaametic egessio models based o ND eos ae povided. The esults obtaied i the pape geealize o impove some coespodig oes fo egatively associated adom vaiables ad egatively othat depedet adom vaiables. Keywods xteded egatively depedet adom vaiables L covegece Macikiewicz Zygmud type momet iequality Law of lage umbes Nopaametic egessio models Mathematics Subject Classificatio 60F05 60F5 60F25 62G05 Suppoted by the Natioal Natual Sciece Foudatio of Chia 50004, 6702), the Natual Sciece Foudatio of Ahui Povice J06) ad the Povicial Natual Sciece Reseach Poject of Ahui Colleges KJ205A08). B Aitig She empess2000@26.com School of Mathematical Scieces, Ahui Uivesity, Hefei 23060, People s Republic of Chia 2 Depatmet of Mathematics ad Statistics, Uivesity of Regia, Regia, SK S4S 0A2, Caada

2 606 A. She, A. Volodi Itoductio Let 2 ad {X, } be a sequece of idepedet adom vaiables with X = 0 ad X < fo all. Bah ad ssee 965) showed that fo ay, X i C X i,.) whee C is a positive costat depedig oly o. The fomula.) is called the -th Bah ssee type momet iequality o Macikiewicz Zygmud type momet iequality. As we kow that the Macikiewicz Zygmud type momet iequality plays a impotat ole i pobability limit theoy ad mathematical statistics, especially i establishig stog covegece, weak covegece ad lage sample popeties of statistics i may stochastic models. Thee ae may sequeces of adom vaiables satisfyig the Macikiewicz Zygmud type momet iequality ude some suitable coditios, such as matigale diffeece sequece see Chatteji 969), ρmixig sequece see Byc ad Smoleski 993 o Wu 2006), egatively associated sequece NA, i shot, see Shao 2000), egatively othat depedet sequece NOD, ishot, seeasadia et al. 2006), egatively supeadditive depedet sequece NSD, i shot, see Hu 2000 o Wag et al. 204), asymptotically almost egatively associated sequece with the mixig coefficiets satisfyig cetai coditios AANA, i shot, see Yua ad A 2009), ad so o. Howeve, thee is o liteatue discussig the Macikiewicz Zygmud type momet iequality fo exteded egatively depedet sequece ND, i shot) which icludes idepedet sequece, NA sequece, NSD sequece ad NOD sequece as special cases. The mai pupose of the pape is to establish the Macikiewicz Zygmud type momet iequality fo ND adom vaiables ad give some applicatios to L covegece, weak ad stog law of lage umbes ude some suitable coditios. I additio, we will peset some applicatios of the L covegece, weak ad stog law of lage umbes to opaametic egessio models based o ND eos. Now, let us ecall the the defiitio of exteded egatively depedet adom vaiables which was itoduced by Liu 2009) as follows. Defiitio. A fiite collectio of adom vaiables X, X 2,...,X is said to be exteded egatively depedet ND, i shot) if thee exists a costat M > 0 such that both ad PX > x, X 2 > x 2,...,X > x ) M PX i > x i ) PX x, X 2 x 2,...,X x ) M PX i x i )

3 Weak ad stog laws of lage umbes fo aays of 607 hold fo all eal umbes x, x 2,...,x. A ifiite sequece {X, } is said to be ND if evey fiite subcollectio is ND. A aay of adom vaiables {X i, i, } is called owwise ND adom vaiables if fo evey, {X i, i } ae ND adom vaiables. If M = i Defiitio., the the ND stuctue educes to the well kow otio of NOD adom vaiables, which was itoduced by Lehma 966)cf.also Joag-Dev ad Poscha 983). The ND stuctue ca eflect ot oly a egatively stuctue but also a positive oe to some extet. Liu 2009) poited out that the ND adom vaiables ca be take as egatively o positively depedet ad povided some iteestig examples to suppot this idea. Joag-Dev ad Poscha 983) also poited out that NA adom vaiables must be NOD ad NOD is ot ecessaily NA, thus NA adom vaiables ae ND. I additio, Chistofides ad Vaggelatou 2004) idicated that NA implies NSD ad Hu 2000) poited out that NSD is NOD. Hece, the class of ND adom vaiables icludes idepedet sequece, NA sequece, NSD sequece ad NOD sequece as special cases. Sice the cocept of ND stuctue was itoduced by Liu 2009), may authos studied the pobability limit popeties fo ND adom vaiables ad povided some iteestig applicatios. See fo example, Liu 200) studied the sufficiet ad ecessay coditios of modeate deviatios fo ND adom vaiables with heavy tails; Che et al. 200) established the Kolmogoov stog law of lage umbes fo ND adom vaiables ad gave applicatios to isk theoy ad eewal theoy; She 20) peseted Rosethal type momet iequality fo ND adom vaiables ad gave some applicatios; Wag ad Wag 203) ivestigated a moe geeal pecise lage deviatio esult fo adom sums of ND eal-valued adom vaiables i the pesece of cosistet vaiatio; Qiu et al. 203), Wag et al. 203), Wag et al. 203), Wag et al. 204), Wu et al. 204) ad Hu et al. 205) povided some esults o complete covegece fo sequeces of ND adom vaiables o aays of owwise ND adom vaiables; Cheg ad Li 204) established the asymptotics fo the tail pobability of adom sums with a heavy-tailed adom umbe ad ND summads; Wag et al. 205) studied the complete cosistecy fo the estimato of opaametic egessio models based o ND eos, ad so foth. Remak. As is metioed i Liu 2009), the ND stuctue ca eflect ot oly a egative depedece stuctue but also a positive oe iequalities fom the defiitio of NOD adom vaiables hold both i evese diectio), to some exted. We efe the iteested eades to xample 4. i Liu 2009) whee ND adom vaiables ca be take as egatively o positively depedet. Hee, we povide two examples possessig the ND stuctue. The fist oe comes fom xample 4.2 i Liu 2009). Fo ay, let X, X 2,...,X be depedet accodig to a copula fuctio Cu, u 2,...,u ) with absolutely cotiuous distibutio fuctios F, F 2,...,F. Assume that the joit copula desity C,2,..., u, u 2,...,u ) = u u 2... u Cu, u 2,...,u )

4 608 A. She, A. Volodi exists ad is uifomly bouded i the whole domai. The adom vaiables {X, } ae ND. As oted i xample 4.2 i Liu 2009), fo example, copulas i the Fak family of the fom C α u, u 2,...,u ) = α l + eαu )...e αu ) ) e α ),α<0 belog to this categoy. The aothe oe comes fom Che et al. 200). Recall that a -dimesioal Falie Gumbel Mogeste FGM, i shot) distibutio has the followig fom ) F,2,..., x, x 2,...,x ) = F k x k ) + i< j a ij F i x i ) F j x j ), whee F k = F k fo k =, 2,..., ae coespodig magial distibutios ad a ij ae eal umbes choose such that F,2,..., x, x 2,...,x ) is a pope - dimesioal distibutio. Che et al. 200) poited out that evey -dimesioal FGM distibutio descibes a specific ND stuctue. The followig cocept of stochastic domiatio will be used i this wok. Defiitio.2 A aay {X i, i, } of adom vaiables is said to be stochastically domiated by a adom vaiable X if thee exists a positive costat C such that fo all x 0, i ad. P X i > x) CP X > x) By usig the cocept of stochastic domiatio, we ca get the followig impotat popety fo stochastic domiatio. Popety. Suppose that the aay {X i, i, } is stochastically domiated by a adom vaiable X. The fo all α>0, thee exists a positive costat C such that X i α C X α fo all i ad. This stuctue of the pape is ogaized as follows: some impotat popeties of ND adom vaiables ae povided i Sect. 2, icludig the Macikiewicz Zygmud type momet iequality ad Rosethal type momet iequality. These popeties will be used to pove the mai esults of the pape. I Sect. 3, some esults o L covegece, weak ad stog law of lage umbes fo aays of owwise ND adom vaiables ae established. Fially, some applicatios of the L covegece, weak ad stog law of lage umbes to opaametic egessio models based o ND eos ae povided i Sect. 4. Thoughout the pape, let C deote a positive costat ot depedig o, which may be diffeet i vaious places. Let I A) be the idicato fuctio of the set A. Deote log x = l maxx, e), x + = xix > 0) ad x = xix < 0).

5 Weak ad stog laws of lage umbes fo aays of Popeties of ND adom vaiables I this sectio, we will peset some impotat popeties of ND adom vaiables icludig the Macikiewicz Zygmud type momet iequality ad Rosethal type momet iequality. These popeties play impotat oles to pove the mai esults of the pape. The fist oe is a basic popety of ND adom vaiables, which ca be foud i Liu 200) fo istace. Lemma 2. Let adom vaiables X, X 2,...,X be ND, f, f 2,..., f be all odeceasig o all oiceasig) fuctios, the adom vaiables f X ), f 2 X 2 ),..., f X ) aealsond. The ext oe is the Rosethal type momet iequality fo ND adom vaiables, which was established by She 20). This iequality with expoet 2 ca be used to pove the Macikiewicz Zygmud type momet iequality. Lemma 2.2 Let {X, } be a sequece of ND adom vaiables with X = 0 ad X p < fo some p 2 ad ay. The thee exist positive costats C p depedig oly o p such that fo ay, p X i C p X i p + X 2 i ) p/2. 2.) With the Rosethal type momet iequality accouted fo, oe ca get the Macikiewicz Zygmud type momet iequality fo ND adom vaiables as follows. The poof is simila to that of Lemma 2. i Che et al. 204). Fo coveiece of the eade, we will peset the poof of Lemma 2.3 i Appedix. Lemma 2.3 Let {X, } be a sequece of ND adom vaiables with X = 0 ad X < fo some 2 ad ay. The thee exist positive costats c depedig oly o such that fo ay, X i c X i. 2.2) Usig Lemma 2.3, we ca get the followig coollay by the same agumet as Theoem 2.3. i Stout 974). Coollay 2. Let {X, } be a sequece of ND adom vaiables with X = 0 ad X < fo some 2 ad ay. The thee exist positive costat c depedig oly o such that fo ay, max j j X i c log X i. 2.3)

6 60 A. She, A. Volodi Remak 2. Assume that 2.2) holds fo ay ad X i coveges almost suely. The we have by Fatou s lemma that X i c X i. 2.4) Remak 2.2 Let {a, } be a sequece of eal umbes. Ude the coditios of Lemma 2.3, wehavefo that a i X i 2 c a i X i. 2.5) Assume futhe that a i X i coveges almost suely, we have by Fatou s lemma that a i X i 2 c a i X i. 2.6) We oly eed to ote that a i = a + i a i, ad fo fixed, {a+ i X i, i } ad {a i X i, i } ae both ND adom vaiables by Lemma Mai esults ad thei poofs I Sect. 2, the Macikiewicz Zygmud type momet iequality fo ND adom vaiables was established. I this sectio, we will give some applicatios of the Macikiewicz Zygmud type momet iequality to L covegece, weak ad stog laws of lage umbes fo aays of owwise ND adom vaiables ude some uifomly itegable coditios. I the followig, let {X i, u i v, } be a aay of adom vaiables defied o a fixed pobability space, F, P),let{u, } ad {v, } be two sequeces of iteges ot ecessay positive o fiite) such that v > u fo all ad v u as.let{k, } be a sequece of positive umbes such that k as ad {h), } be a iceasig sequece of positive costats with h) as. 3. L covegece ad weak law of lage umbes The otio of h-itegability with expoet was itoduced by Sug et al. 2008), which deals with usual omed sums of adom vaiables as follows. Defiitio 3. Let {X i, u i v, } be a aay of adom vaiables ad > 0. The aay {X i, u i v, } is said to be h-itegable with expoet if

7 Weak ad stog laws of lage umbes fo aays of 6 sup k v i=u X i < ad lim k v i=u X i I X i > h)) = 0. Ude the coditios of h-itegability with expoet, Sug et al. 2008) futhe studied the L covegece ad weak law of lage umbes fo aays of owwise NA adom vaiables. Ispied by the cocept of h-itegability with expoet, Wag ad Hu 204) itoduced a ew ad weake cocept of uifom itegability as follows. Defiitio 3.2 Let {X i, u i v, } be a aay of adom vaiables ad > 0. The aay {X i, u i v, } is said to be esidually h-itegable R-h-itegable, i shot) with expoet if sup k v i=u X i < ad lim v k i=u X i h / )) I Xi > h)) = 0. Ude the coditio of R-h-itegability with expoet, Wag ad Hu 204) established some weak laws of lage umbes fo aays of depedet adom vaiables. Notig that X i h )) / I Xi > h)) X i I X i > h)), hece, the cocept of R-h-itegability with expoet is weake tha h-itegability with expoet. Fo moe details about the L covegece ad weak law of lage umbes fo omed sums o weighted sums of adom vaiables based o h-itegability o R-hitegability, oe ca efe to Yua ad Tao 2008), Odóñez et al. 202), She et al. 203), Sug 203), ad so o. Ispied by Wag ad Hu 204) ad Sug 203), we get the followig esults o L covegece ad weak law of lage umbes fo aays of owwise ND adom vaiables. the fist oe deals with the L covegece ad weak law of lage umbes fo omed sums of aays of owwise ND adom vaiables. Theoem 3. Suppose that {X i, u i v, } is a aay of owwise ND R-h-itegable with expoet < 2 adom vaiables. Let k,h), ad h)/k 0 as. The k / v i=u X i X i ) 0 i L ad, hece, i pobability as.

8 62 A. She, A. Volodi Poof Fo fixed, deote fo u i v that ) Y i = h / )I X i < h / ) + X i I ) + h / )I X i > h / ), ) Z i = X i Y i = X i + h / ) I ) ) + X i h / ) I X i > h / ), Notig that S = k / v Y i Y i ), T = i=u k / ) X i h / ) ) X i < h / ) v i=u Z i Z i ). k / v i=u X i X i ) = S + T,, we have by C -iequality that k / v X i X i ) i=u C S + C T. To pove v k / i=u X i X i ) 0iL, we oly eed to show S 0 ad T 0as, whee < 2. Fistly, we will show that S 0as. Note that < 2, it suffices to show S 2 0as. Fo fixed, it is easily checked that {Y i Y i, u i v } ae ND adom vaiables by Lemma 2.. Notig that < 2 ad Y i = mi{ X i, h / )},we have by Lemma 2.3 o Remak 2. that S 2 = C k 2/ C k 2/ k / v v Y i Y i ) i=u Y i Y i ) 2 C i=u k 2/ h 2 )/ ) [ ] h) 2 )/ C k 0 as, k v 2 i=u Y i v i=u X i v i=u Y 2 i

9 Weak ad stog laws of lage umbes fo aays of 63 which implies that S 2 0as ad thus, S 0as. Next, we will show that T 0as. Fo fixed, we ca see that {Z i Z i, u i v } ae still ND adom vaiables by Lemma 2. agai. Notig that Z i = ) ) X i h / ) I X i > h / ), we have by Lemma 2.3 o Remak 2. agai that T = k / v v Z i Z i ) i=u C Z i Z i k i=u C k v i=u 0 as, X i h / ) v C Z i k i=u ) I Xi > h) ) which implies that T 0as. This completes the poof of the theoem. Remak 3. Note that the cocept of R-h-itegability with expoet is weake tha h-itegability with expoet ad ND is weake tha NA. Hece, the esult of Theoem 3. geealizes ad impoves the coespodig oe of Sug et al. 2008) fo NA adom vaiables to the case of ND adom vaiables. I additio, the esult of Theoem 3. geealizes the coespodig oe of Wag ad Hu 204) fo NOD adom vaiables to the case of ND adom vaiables. The ext oe deals with the L covegece ad weak law of lage umbes fo weighted sums of ND adom vaiables. The poof is simila to that of Theoem 2. i Sug 203). So the details ae omitted. We should poit out that the key techique used hee is still the Macikiewicz Zygmud type momet iequality fo ND adom vaiables. Theoem 3.2 Let < 2, {X i, u i v, } be a aay of owwise ND adom vaiables ad {a i, u i v, } be a aay of costats. Assume that the followig coditios hold: i) sup v i=u a i X i < ; ii) fo ay ɛ>0, lim v i=u a i X i I X i >ɛ)= 0.

10 64 A. She, A. Volodi The v i=u a i X i X i ) 0 i L ad, hece, i pobability as. With Theoem 3.2 accouted fo, we ca get the followig coollay. The poof is simila to that of Coollay 2. i Sug 203), so the details ae omitted. Coollay 3. Let {a i, u i v, } be a aay of costats satisfyig. k = / supu i v a i, 0 < h) ad h)/k 0 as. Let {X i, u i v, } be a aay of owwise ND h-itegable with expoet < 2 adom vaiables. The v i=u a i X i X i ) 0 i L ad, hece, i pobability as. If we take a i = k / fo u i v ad i Coollay 3., the we ca get the followig coollay. Coollay 3.2 Let {X i, u i v, } be a aay of owwise ND h- itegable with expoet < 2 adom vaiables, k, 0 < h) ad h)/k 0 as. The v i=u X i X i ) k / 0 i L ad, hece, i pobability as.. Remak 3.2 Note that the coditio k = / supu i v a i,0< h) ad h)/k 0as i Coollay 3. i the pape is weake tha. k = / supu i v a i,0< h) ad h)/k 0as i Coollay 3.6 of Wag ad Hu 204). Hece, ou esults of Theoem 3.2 ad Coollay 3. geealize ad impove the coespodig oe of Coollay 3.6 i Wag ad Hu 204) fo NOD adom vaiables to the case of ND adom vaiables. 3.2 Stog law of lage umbes I ode to establish the stog vesio of Theoem 3., we should itoduce the cocept of stogly esidual h-itegability with expoet, which deals with usual omed sums of adom vaiables as follows.

11 Weak ad stog laws of lage umbes fo aays of 65 Defiitio 3.3 Let {X i, u i v, } be a aay of adom vaiables ad > 0. The aay {X i, u i v, } is said to be stogly esidually h-itegable SR-h-itegable, fo shot) with expoet if ad v k = i=u sup k v i=u X i < X i h / )) I Xi > h) ) <. Remak 3.3 We poit out that the cocept of S R h-itegability with expoet is stoge tha the cocept of R h-itegability with expoet. S R h-itegability with expoet implies R h-itegability with expoet. Ou mai esult o the stog law of lage umbes fo usual omed sums of aays of owwise ND adom vaiables is as follows. Theoem 3.3 Suppose that {X i, u i v, } is a aay of owwise ND S R-h-itegable with expoet < 2 adom vaiables. Let k,h), ad ) 2 h) k <. The v X = k / i X i ) 0 a.s. as. i=u Poof We use the same otatios as those i Theoem 3.. I ode to pove k / ad v i=u X i X i ) 0 a.s. as, we oly eed to show S. = k / T. = k / v i=u Y i Y i ) 0 a.s. as, 3.) v i=u Z i Z i ) 0 a.s. as. 3.2) Fo 3.), otig that < 2 ad Y i = mi{ X i, h / )}, wehaveby Makov s iequality, Lemma 2.3 o Remak 2. that fo ay ε>0, P S >ε) ε 2 = C = = k 2/ k 2/ v Y i Y i ) i=u v i=u Y 2 i 2

12 66 A. She, A. Volodi C C <, = h 2 )/ ) k 2/ h) = k ) 2 v Y i i=u v sup X i k i=u which togethe with Boel Catelli lemma implies 3.). Fo 3.2), otig that Z i = X i h / ))I X i > h)), wehaveby Makov s iequality, Lemma 2.3 o Remak 2. agai that fo ay ε>0, P T >ε) ε = C = C <, = k v Z i Z i ) i=u v k = v k = i=u i=u Z i ) X i h / ) I Xi > h) ) which togethe with Boel Catelli lemma yields 3.2). This completes the poof of the theoem. Usig Theoem 3.3, we ca get the followig stog law of lage umbes fo weighted sums of aays of owwise ND adom vaiables. Coollay 3.3 Let < 2, {X i, u i v, } be a aay of owwise ND adom vaiables ad {a i, u i v, } be a aay of costats. Let h), ad v i) sup a i X i < ; i=u v ii) a i X i I X i > h)) < ; = i=u ) 2 iii) h) sup a i <. = u i v The v i=u a i X i X i ) 0 a.s. as. 3.3)

13 Weak ad stog laws of lage umbes fo aays of 67 Poof Deote k = / sup a i. It follows by coditio iii) that h)/k 0 u i v as, ad thus k as. Without loss of geeality, we assume that a i 0 fo all u i v ad. Othewise, we will use a i + ad ai istead of a i espectively ad ote that a i = a i + a i. Hece, it follows by Lemma 2. that {k/ a i X i, u i v, } is still a aay of owwise ND adom vaiables. Takig k / a i X i istead of X i i Theoem 3.3, we have by coditio i) that sup Notig that that k / v k / k i=u a i X i = sup v i=u a i X i <. 3.4) a i fo all u i v ad, we have by coditio ii) v k = i=u v k = i=u v ) k / a i X i h / ) I k / a i X i I k / a i X i I Xi > h) ) = i=u <. k / ) a i X i > h) ) a i X i > h) 3.5) Hece, the desied esult 3.3) follows by 3.4), 3.5) ad Theoem 3.3 immediately. The poof is completed. Remak 3.4 Accodig to the poofs of Theoem 3.3 ad Coollay 3.3, oe ca get the complete covegece fo aays of owwise ND adom vaiables, which is much stoge tha almost sue covegece. Ude the coditios of Theoem 3.3,we have that fo ay ε>0, P = k / v X i X i ) i=u >ε < ; 3.6) ude the coditios of Coollay 3.3, we have that fo ay ε>0, v P a i X i X i ) i=u >ε <. 3.7) =

14 68 A. She, A. Volodi 4 Applicatios I Sect. 3, we established the L covegece, weak ad stog laws of lage umbes fo aays of owwise ND adom vaiables ude some uifomly itegable coditios. I this sectio, we will peset some applicatios of the L covegece, weak ad stog laws of lage umbes to opaametic egessio models based o ND eos. Coside the followig opaametic egessio model: Y k = gx k ) + ε k, k =, 2,...,,, 4.) whee x k ae kow fixed desig poits fom A, ad A R m is a give compact set fo some m, g ) is a ukow egessio fuctio defied o A, ad the ε k ae adom eos. As a estimato of g ), we coside the weighted egessio estimato as follows: g x) = W k x)y k, x A R m, 4.2) whee W k x) = W k x; x, x 2,...,x ), k =, 2,..., ae the weight fuctios. The above weighted egessio estimato fo opaametic egessio model was fist adapted by Geogiev 985). Sice the, may authos devoted to studyig the asymptotic popeties of g x) ad povidig may iteestig esults. We efe the eades to Roussas 989), Fa 990), Roussas et al. 992), Ta et al. 996), Liag ad Jig 2005), Wag et al. 204), Wag ad Si 205), Che et al. 206) fo istace. The pupose of this sectio is to futhe ivestigate the stog cosistecy ad mea cosistecy fo the estimato g x) i the opaametic egessio model based o ND eos by usig the esults obtaied i Sect. 3. I this sectio, let cg) deote the set of cotiuity poits of the fuctio g o A. The symbol x deotes the uclidea om. Fo ay fixed desig poit x A, the followig assumptios o weight fuctios W k x) will be used: H ) H 2 ) H 3 ) W k x) as ; W k x) C < fo all ; W k x) gx k ) gx) I x k x >a) 0as fo all a > 0. We poit out that the desig assumptios H ) H 3 ) ae egula coditios fo opaametic egessio models ad ae vey geeal. Fo moe details, oe ca efe to Liag ad Jig 2005) ad Wag et al. 204) fo istace. Based o the assumptios

15 Weak ad stog laws of lage umbes fo aays of 69 above, we peset the followig esults o stog cosistecy ad mea cosistecy of the opaametic egessio estimato g x). The fist oe is the stog cosistecy of the opaametic egessio estimato g x) Theoem 4. Let {ε k, k, } be a aay of owwise ND adom vaiables with mea zeo which is stochastically domiated by a adom vaiable X with X < fo some < < 2. Suppose that the coditios H ) H 3 ) hold, ad { } max W kx) =O u ) fo some u > max k 2,. 4.3) The fo all x cg), g x) gx) a.s. 4.4) Poof Fo a > 0 ad x cg), we obtai fom 4.) ad 4.2) that g x) gx) W k x) gx k ) gx) I x k x a) + W k x) gx k ) gx) I x k x > a) + gx) W k x). 4.5) It follows fom x cg) that fo all ε>0, thee exists a costat δ>0 such that fo all x which satisfy x x <δ,wehave gx ) gx) <ε. Hece we take 0 < a <δi 4.5) ad obtai that g x) gx) ε W k x) + W k x) gx k ) gx) I x k x > a) + gx) W k x). The by assumptios H )-H 3 ) ad the abitaiess of ε>0, we have that fo all x cg), lim g x) = gx). 4.6)

16 620 A. She, A. Volodi Hece, to pove 4.4), it suffices to pove g x) g x) = W k x)ε k 0 a.s. 4.7) We will apply Coollay 3.3 with X k ) = ε k, a k = W k x), u =, v = ad h) = a, whee 0 < a < u 2.By X <, coditios H 2 ),4.3) ad Popety., wehavethat sup v i=u a i X i C sup ) max W kx) k C sup u ) C <, W k x) X v a i X i I Xi > h) ) C W k x) X = i=u = C u ) <, = h) sup a i u i v = ) 2 C a u)2 )/ <. = Thus, the coditios i) iii) i Coollay 3.3 ae satisfied. Notig that ε k = 0, we ca immediately get the desied esult 4.7) by Coollay 3.3. This completes the poof of the theoem. The ext oe is the mea cosistecy ad weak cosistecy of the opaametic egessio estimato g x). Theoem 4.2 Let {ε k, k, } be a aay of owwise ND adom vaiables with mea zeo which is stochastically domiated by a adom vaiable X with X < fo some < < 2. Suppose that the coditios H ) H 3 ) hold, ad The fo all x cg), max W kx) =O u ) fo some u > ) k g x) gx) i L, 4.9)

17 Weak ad stog laws of lage umbes fo aays of 62 ad thus, g x) gx) i pobability. 4.0) Poof Simila to the poof of Theoem 4., we ca see that 4.6) still holds. Note that g x) gx) 2 g x) g x) + 2 g x) gx). Hece, to pove 4.9), it suffices to pove g x) g x) = W k x)ε k 0 as. 4.) We will apply Theoem 3.2 with X k = ε k, a k = W k x), u = ad v =. By X <, coditios H 2 ),4.8) ad Popety., we have that sup v ad fo ay ɛ>0, i=u a i X i C sup v ) max W kx) k C sup u ) C <, W k x) X i=u a i X i I X i >ɛ) C u ) 0 as. Thus, the coditios i) ad ii) i Theoem 3.2 ae satisfied. Notig that ε k = 0, we ca immediately get the desied esult 4.) by Theoem 3.2. This completes the poof of the theoem. Ackowledgemets The authos ae most gateful to aoymous efeees fo caeful eadig of the mauscipt ad valuable suggestios which helped i impovig a ealie vesio of this pape. Appedix Poof of Lemma 2.3. If = o = 2, the we ca see that 2.2) holds tivially by C -iequality ad Lemma 2.2 with p = 2, espectively. So i the followig, we oly eed to coside the case < < 2. Fo fixed, deote M = X i. Without loss of geeality, we assume that M > 0. Fo ay ε>, it is easily see that X i + ε)m + P +ε)m ) X i > t / dt. 4.2)

18 622 A. She, A. Volodi Fo fixed ad t + ε)m, deote fo i that Y i = t / I X i < t / ) + X i I X i t / ) + t / I X i > t / ). It follows by 4.2) that X i Fo I,wehave I + ε)m + +ε)m ) P X i > t / dt ) + P Y i > t / dt +ε)m ) + ε)m + P X i > t / dt +ε)m ) + P Y i Y i ) +ε)m > t/ Y i dt. = + ε)m + I + I ) 0 ) P X i > t / dt = X i = M. 4.4) Note that sup t / Y t +ε)m i 2 sup t / t / X i I X i > t / ) t +ε)m 2 + ε). 4.5) Hece, by 4.5), Makov s iequality ad Lemma 2.2 with p = 2, we ca get that ) [ I 2 P Y +ε)m i Y i ) > 2 + ε) ] t / dt [ 2 + ε) ] 2 t 2/ 2 Y +ε)m i Y i ) dt [ C ε) ] 2 [ + C ε) ] 2 +ε)m t 2/ X 2 i I X i t / )dt +ε)m P X i > t / )dt

19 Weak ad stog laws of lage umbes fo aays of 623. = I 2 + I ) Hee, C 2 is defied by Lemma 2.2. FoI 22,wehave [ I 22 C ε) ] 2 0 P X i > t / )dt = C 2 [ 2 + ε) ] 2 M. 4.7) Fo I 2, we ca see that +ε)m t 2/ X 2 i I X i t / )dt P X i > y /2 )dy +. = J + J 2. +ε)m t 2/ dt +ε)m t 2/ dt t 2/ +ε) 2/ M 2/ +ε) 2/ M 2/ 0 P X i > y /2 )dy 4.8) Fo J, it follows by Makov s iequality that Fo J 2,wehave +ε) 2/ M 2/ J 2 + ε) 2/ M 2/ X i y /2 dy 0 2 = 2 ) 2 X i. 4.9) J 2 = +ε) 2/ M 2/ = 2 2 +ε) 2/ M 2/ Hece, by 4.6)-4.20), we ca get that 0 P X i > y /2 )dy t 2/ dt y /2 y /2 P X i > y /2 )dy y /2 P X i > y /2 )dy = 2 2 X i. 4.20) [ I 2 C ε) ] 2 [ M + C ε) ] [ ) [ = C 2 [ 2 + ε) ] 2 + ] M 2 ) ] 2 2 M. 4.2) 2

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