Complete Convergence for Weighted Sums of Arrays of

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1 Thailad Statisticia Jauary 01; 10(1 : htt://statassoc.or.th Cotributed aer Comlete Coergece for Weighted Sums of Arrays of Rowwise ~ -mixig Radom Variables Kamo Budsaba* [a,b] Qiu Dehua [c] Hear Urmeeta [d] Adrei Volodi [e] [a] Deartmet of Mathematics ad Statistics, Thammasat Uersity Ragsit Ceter, Pathum Tha, 111, Thailad. [b] Cetre of Excellece i Mathematics, CHE, Si Ayutthaya Rd. Bago, 10400, Thailad. [c] School of Mathematics ad Comutatioal Sciece, Guagdog Uersity of Busiess Studies, Guagzhou 51030, P.R.Chia. [d] Deartameto de Estadística e Iestigació Oeratia de la Uersidad Pública de Naarra, Pamloa, Sai. [e] Deartmet of Mathematics ad Statistics, Uersity of Regia, Regia, S4S 0A, Caada. * Author for corresodece; amo@mathstat.sci.tu.ac.th Receied: 0 March 011 Acceted: 19 May 011 Abstract I this aer we obtai some ew results o comlete coergece for weighted sums of arrays of rowwise ~ -mixig radom ariables. Our results imroe ad exted the some results established for sequeces of ideedet radom ariables. Keywords: comlete coergece, ~ -mixig radom ariables, slowly aryig fuctio, weighted sums.

2 14 Thailad Statisticia, 01; 10(1: Itroductio The cocet of comlete coergece was itroduced by Hsu ad Robbis i [1] as follows: A sequece of radom ariables {, 1} are said to coerge comletely to a costat C if =1 P ( C> < for all > 0 o, may authors hae deoted their study to comlete coergece.. From the Recetly, Sug [] roed the followig two results. I Theorems A ad B we assume that {, 1} is a sequece of zero-mea ideedet radom ariables stochastically domiated by a radom ariable, that is, P( > x CP( > x x ad all 1 ad some ositie costat C. Moreoer, for all > 0 { a,i 1, 1} is a array of real umbers satisfyig such that su 1,i1 a < ad a i is fite almost surely for all 1. Fially, let t 1, < <, >0 be costats such that = (t1 > 0. Theorem A. Assume that E < ad (i If 1 <, or (ii if, ad the a = O( for some <. 1 q a = O( for some q < /, ( t 1/ P( a i > < for all > 0. (3 =1 Theorem B. Assume that E log < ad (i If 1 < or a = O( for all 1. (4

3 Kamo Budsaba 143 q ad a = O( for some q < /, (ii If the (3 holds. Let Z be the set of itegers ad { a, 1, Z} costats. Deote be a array of N(,m 1 =#{Z : a (m 1 },, 1,m 1, where the symbol # A stads for the umber of elemets i the set A. For two sequeces of real umbers { a m,m 1} ad { b m,m 1}, we write am bm as m, if am = O(bm ad ise ersa bm = O(am as m. Wag et al. [3] roed the followig result: Theorem C. Let r > 1 ad { i,i Z} be a sequece of i.i.d. radom ariables ad let { a, 1,iZ} for be a array of costats. (I If > ad N m m as m whe q q( r1/ (, 1, 1,, <, (5 E = 0, whe 1 q( r 1, (6 iz a = O(,, whe q( r 1, for some 0 < < /, (7 the the followig statemets are equialet: (ie (r1 < ; i(r1 ( ii max P a i 1 i < Z = q = (II If ad N(,m 1 m (r1 E =0, whe 1 (r 1,, 1, as m, > < for all > 0.

4 144 Thailad Statisticia, 01; 10(1: a iz (r1 the the followig statemets are equialet: (ie (r1 log(1 = O(1,, < ; i(r1 1/ ( ii max P a i 1 i < Z > <, for all > 0. The mai urose of this aer is to geeralize the aboe metioed results for ~ -mixig radom ariables (see the defitio below. Theorem A ad Theorem B ad the sufficiet art of Theorem C are exteded ad imroed for ~ -mixig case. Let {,,P} be a robability sace. I the followig, all radom ariables are assumed to be defied o {,,P}. For a sequece of radom ariables {, 1} we deote ( : S N S =. Gie two -subalgebras 1,, deote ( 1, =su{ corr(,, L( 1, L ( }, where the correlatio coefficiet is defied i usual way E( EE corr(, = Var( Var( ad by L ( we deote the sace of all -measurable radom ariables such that E( <. Stei [4] itroduced the followig coefficiets of deedece (with slightly differet otatios: ~ ( =su{ ( S, T : all fite subsets S,T N such that dist(s,t }, 0. Obiously, 0 ~ ( 1 ~ ( 1, 0, ad ~ (0 = 1.

5 Kamo Budsaba 145 Defitio. A sequece of radom ariables, 1} sequece if there exists { are said to be a ~ -mixig N such that ~ ( < 1. A array of radom ariables {, 1, 1} are said to be a array of rowwise ~ -mixig radom ariables, if, for eery ositie iteger the sequece of radom ariables {, 1} is a ~ - mixig sequece. For fixed -th row of a array of rowwise ~ -mixig radom ariables {, 1, 1} {, 1} as ~ ( for eery 1. we deote the coefficiets of deedece of the sequece The otio of ~ -mixig assumtio is similar to -mixig, but they are quite differet from each other. A umber of ublicatios are deoted to ~ -mixig sequece. We refer to Bradley [5,6] for the cetral limit theorem, Bryc ad Smolesi [7] for momet iequalities ad almost sure coergece, Shachao [8] for momet iequalities ad strog law of large umbers, Gut ad Peligrad [9], Wu [10,11], ad Shixi [1] for almost sure coergece, Ute ad Peligrad [13] for maximal iequalities ad the iariace ricile, Dehua ad Shixi [14,15] for comlete coergece, Dehua ad Shixi [16] for Háec-Rèyi iequality ad strog law of large umbers amog may others. > 0 Recall that a measurable fuctio h is said to be slowly aryig if for each lim x h( x =1. h(x We refer to Seeta [17] for other equialet defitios ad for detailed ad comrehesie study of roerties of such fuctios. a Throughout this aer, we assume that is fite almost surely, C is a ositie costat which may ary from oe lace to aother, the symbol [ x] deotes the greatest iteger less tha x, ad the symbol iteger more tha x. x deotes the least

6 146 Thailad Statisticia, 01; 10(1: Lemmata I order to roe our mai result, we eed the followig lemmas. The roof of the first lemma could be foud i Ute ad Peligrad [13]. Lemma 1. For a ositie iteger J ad 0 r < 1 ad u, there exists a ositie costat C =C(u,J,r such that if {, 1} is a sequece of radom ariables ~ u with (J r,e = 0, ad E <, the for all 1, for eery 1 u i u E max CE E. 1i =1 =1 =1 The secod lemma is well ow ad we do ot reset the roof. Lemma. Let {, 1, 1} be a array of radom ariables stochastically domiated by a radom ariable, the there exists a costat D such that for all u >0 ad x > 0, (i u u u E I( x D{E I( x x P( > x}, u u E I( > x DE I( > x (ii. u/ The roof of the last lemma could be foud i Bai ad Su [18] Lemma 3. Let (x > 0, the h be a slowly aryig fuctio as x h(x lim su = x 1 h( lim x h(x =, lim x h(x x x > 0, > (i 1, (ii For all 0, ad all ositie itegers C h( =1 h( C = 0, for all h( > 0..

7 Kamo Budsaba 147 (iii For all < 0, > 0 all ositie itegers C 3. Mai Results ad Proofs h( = h( C h(. With the relimiaries accouted for, we ca ow formulate ad roe mai results of this aer. Theorem 1. Let > 0, t, be costats such that t > 1, (x > 0 aryig fuctio, {, 1, 1} h be a slowly be a array of zero-mea rowwise ~ -mixig radom ariables stochastically domiated by a radom ariable { a,i 1, 1} be a array of costats satisfyig (1. Assume that ~ lim su ( = (t1 > where 0. If = 1 (i If = 1 <1 ad E t we additioally assume that, ad E <, the h( <, E <. ( t1 1/ h( max P a > < for all > 0, (8 < 1 =0 moreoer t 1/ h( P a > < for all > 0. (9 =1 (ii If < < 1, the (8 ad (9 hold. (iii If =,{a, 1,i 1} satisfies (, ad E < ad { a, 1,i 1} (i If satisfies (, the (8 ad (9 hold., ad, the (8 ad (9 hold. Lemma 3 we hae Proof. First of all we ote that it is eough to show that (8 holds. Really, by

8 148 Thailad Statisticia, 01; 10(1: CC =0 =1 (t1 therefore, (9 holds. If < 1 t 1 t h(p a > t = h(p a 1 =0 < h( max < 1 P a > > <, t, the by Lemma 3 (i we obtai that (8 holds. Thus, we assume that. Sice a ositie iteger is fite almost surely for each 1, there exists such that P( a i= 1 > / < By Lemma 3 (iii, i order to roe (8, it is eough to show that t, for all ( t1 1/ h( max P( a > / <. (10 < 1 =0 1. Without loss of geerality, we assume that a > 0 for all 1,i 1,sui1,1a =1 a For i 1, 1 we defie, ad i =1 a. Thus, for ay 0. (11, we hae U = I(a,V = I(a >. Sice E = 0, we obtai

9 Kamo Budsaba 149 (t1 (t1 < 1 h( max P( a =0 < 1 h( max P( a (U EU > /4 =0 (t1 < 1 h( max P( a (V EV> /4 =0 Def J 1 J > / =. (1 We estimate each term 1 J ad For J, we first roe that 1/ aev as If = 1 J searately. 0. (13, sice E < E, by Lemma ad (11, we hae 1/ aev (t1 If > 1, t > 1 E I( > E I( > 0 as., select such that max {,,1}< < h( < Lemma ad (11, we obtai. Sice, the by Lemma 3 (i, we hae E <. Therefore, by 1/ ( a EV E a / E I( > I( a >

10 150 Thailad Statisticia, 01; 10(1: If > 1, t = 1 (/ = E I( > 0 as., sice < E we obtai 1/ ( a EV E a / E I(> = E I(> I( a 0 as. Thus, (13 holds. Hece, there exists large eough such that 1/ aev > < / 8. (14 Select > 0 such that > 0 ad >, by (14, (11, Lemma (ii ad Lemma 3 (ii,we hae (t1 < 1 J C h( max P( a V > /8 =0 C =0 C =0 (t1 C h( max P( a > =0 < 1 (t1 ( / h( max E a I( a 1 < (t1 h( ( / = C =0 / E h( E I( > I( > / / >

11 Kamo Budsaba 151 = C =0 / = C E h( E i= I( i/ < I( i/ (i < i i=0 =0 i i/ (i CC h( E I( < i=0 (i / h( C CE h( <. (15 I order to estimate J 1, we first ote that obiously for eery ositie iteger, { U EU,1i} is a sequece of zero-mea ~ -mixig radom ariables with the mixig coefficiet ot greater tha ~ (. Fix ay ad > (the alue of will be secified later. By Maro's iequality, Lemma 1, ad Cr -iequality,, we hae (t1 / J1 C h( max E a 1 U ( E au =0 < / C =0 (t1 h( max < 1 / E a U ( E a U / Def J 3 J 4 =. (16 Let I ={i: ( 1 < a }, 1, 1, the = N for all 1. Sice For > 0 (/ (/ =1 I > >, we hae > for all >,, 1.

12 15 Thailad Statisticia, 01; 10(1: / a = a (#I( 1 =1i I =1 / (/ ( 1 ( 1 = / / (/ > (#I. = (#I For < 0, we also hae Therefore, a > (#I = = =1i I / (/ a. (#I =1 / / ( / (# I C for all 1. (17 = By the same way as we roed (15 ad by Lemma (i, we hae J 3 C C =0 =0 ( t1 (t1 h( C h( max < 1 =0 ( t1 h( max < 1 =1 (#I max P( a < 1 ( > P( a / 1/ > E E a 1/ I( < (( 1 I( a 1/

13 Kamo Budsaba 153 = C C =0 ( t1 h( max < 1 =1 (# I ( ( 1 E I(( i 1 1/ < i 1/ = C C C =0 =0 ( t1 ( t1 h( h( max < 1 =1 max 1 < = (# I (# I ( ( ( 1 i=1 E E I(( i 1 I(( i 1 1/ 1/ < i < i 1/ 1/ Def = C J J. (18 Sice 5 6 >, we hae that ( / < 0 (t1. The by (17 ad Lemma 3 < 1 ( / / J5 C h( max E I((i1 < i =0 CCE i=5 C =0 C =1 I((i1 ( / 4 h( E h( E < i i=5 I((i1 = log i I((i1 < i ( / < i h( ( / CC i h(ie I((i1 < i i=5 C CE h( <. (19 Next,

14 154 Thailad Statisticia, 01; 10(1: (t1 / J6 C h( max (#I( E I((i1 < i =0 < 1 i=1 i =[ 1] (t1 / i (/ C h( max ( E I((i1 < i =0 < 1 i=1 (/ C i E I((i1 < i i=3 (t1 ( / (/ C h( i E I((i1 < i =1 i= [log i] (/ ( / CE Ci E I((i1 < i h( i= =1 (/ CC i h(ie I((i1 < i i= C CE h( <. (0 Therefore, from (18, (19, ad (0 we hae that For 4 J, if by ( we hae J 3 < for 1. q E au C E a C. (1 Sice q < /, we ca chose large eough such that ( t 1 (q/ < 0 By Lemma 3 (iii we obtai 4. ( t1 / q/ {( t1 ( q/1/ } ( max ( <. ( < 1 =0 =0 J C h C h If 1 <, let =, the = J < (10 holds. J4 3. Therefore < J 1 for 1. By (1,

15 Kamo Budsaba 155 Remar 1. (i If there exists a ositie costat M > 0 such that h(x M for sufficietly large x, the the assumtio E (t1 h( < imlies that (t1 E <. (ii Let h(x =1, = i, for all i 1, 1, ad { i,i 1} be a sequece of ideedet radom ariables. The Theorem A follows from Theorem 1, sice ideedet radom ariables are a secial case of ~ -mixig radom ariables., ad h (x = 1 (iii Let = 0, t = r accordig to (.11 of Wag et al. [], with = Whe < q(r 1 <. If coditio (5 holds, the (1 holds q ~ ( r 1, = (r 1, q < q ~ 0, by (.11 of Wag et al. [], we hae that = O(1 Therefore, if (5 ad (7 hold, we hae = O( i a <. i a Z. Z, for 0 < < /. Thus Theorem 1 exteds ad imroes the sufficiet art of Theorem C (I for the case of ~ - mixig radom ariables. If coditio (1 o the weights is relaced by a weaer coditio (4, we obtai the followig theorem. Theorem. Let {, 1, 1} be a array of zero-mea rowwise ~ -mixig radom ariables stochastically domiated by a radom ariable. Assume that su ~ lim ( <1 ad E log <, where = (t1 > 0 ad > 0. Let { a,i 1, 1} be a array of real umbers satisfyig (4. (i If 1 <, the ( t1 1/ max P a > < for all > 0, (3 < 1 =0 moreoer t 1/ P a for all =1 > < > 0. (4

16 156 Thailad Statisticia, 01; 10(1: (ii If ad { a,i 1, 1} satisfies (, the (3 ad (4 hold. Proof. Let U,V,I, J be as i the roof of Theorem 1. From this roof, it is sufficiet to show Sice J < ad J <, = 4,5,6 with (x = 1 For J, we first roe that a EV 0 as. E log <, we hae < h. E ad hece 1/ ( a EV E a / E I(> = (t1 E I(> Therefore, there exists large eough such that Thus, similar to the roof of (15 C =0 I( a 0 as. a (t1 =0 < 1 (t1 / max E a < 1 (t1 / EV J C max P( a CC =0 I( a E > < /8. > > I( > /

17 Kamo Budsaba 157 Sice Hece > = C CE =0 = CC E I( > I( i/ / < (i =0i= i/ (i = CC ie I( < i=0 CCE log <., we hae = a (#I > = / = = ( 1 (#I =1i I / a ( 1 / (/ (#I =1 (/. ( 1 / / ( / (# I C for all 1. (5 = By (5, similar to the roof of (19, we obtai (t1 / < 1 ( / CC i E I((i1 < i i=5 J5 C max E I((i1 < i =0

18 158 Thailad Statisticia, 01; 10(1: CCE <. By (5, similar to the roof of (0, we obtai (/ J6 C i E I((i1 < i (t1 ( / (/ C i E I((i1 < i =1 i= (/ CE C i E I((i1 < i (/ CC i E I((i1 < i i= CCE <. Similar to the roof of Theorem 1, we hae J 4 <. Remar. Obiously, Theorem B follows from Theorem by let h(x =1, =, for all i 1, 1, ad { i,i 1} be a sequece of i ideedet radom ariables. Furthermore, Theorem exteds ad imroes the sufficiecy art of Theorem C (II for the case of ~ -mixig radom ariables. Corollary 1. Let {, 1, 1} be a array of zero-mea rowwise ~ -mixig radom ariables stochastically domiated by a radom ariable. Assume that su ~ lim ( <1 ad E < { a,i 1, 1} be a array of real umbers satisfyig ( ad a = O(1 for some >. Let for some <.

19 Kamo Budsaba 159 The ad max P a 1 =0 < > < for all > 0 P( a > < for all > 0. =1 Proof. Let t = 0 ad = 0 ad h (x = 1. Clearly a = O(1. Thus the result follows from Theorem 1 (iii. Corollary. Let {, 1, 1} be a array of zero-mea rowwise ~ -mixig radom ariables stochastically domiated by a radom ariable. Assume that su ~ lim ( <1 ad E log <. Let { a,i 1, 1} array of real umbers satisfyig The ad Proof. Let = 0, = 0 Theorem (ii. a = O(1 max P a 1 =0 < > < be a for all > 0 1/ P( a > < for all > 0. =1 t, ad =. Clearly a = O(1. Thus the result follows from

20 160 Thailad Statisticia, 01; 10(1: Remar 3. Set = for all 1 ad i 1, let { i,i 1} be a sequece of i i.i.d. radom ariables. I this articular case Corollaries 1 ad were roed by Li et al. [19]. Hece Corollaries 1 ad exted the results of Li et al. [19]. As a corollary of Theorem 1, we ca obtai the followig result o the rate of coergece for moig aerage rocesses. Corollary 3. Let {, Z,Z} be a array of zero-mea rowwise ~ -mixig radom ariables stochastically domiated by a radom ariable. Assume that ~ lim su ( <1 ad E (t < (t >1. Let { a, < < } for some 0 < < ad be a sequece of real umbers such that = a <. Set a i = =i 1a for each i ad. The ad (t1 max P a 1 =0 < i= > < t P( a / > < =1 i= for all for all > 0, > 0. Proof. Reeats the roof of Sug [1] ad hece omitted. Remar 4. Corollary 3 exteds Corollary 3 of Sug [] for arrays of rowwise ~ -mixig radom ariables. Refereces [1] Hsu, P. L., Robbis H., Comlete coergece ad the law of large umbers. Proc. Nat. Acad. Sci. USA., 1947; 33: [] Sug, S. H., Comlete coergece for weighted sums of radom ariables. Statist. Probab. Lett., 007; 77: [3] Wag, Y. B., Liu,. G., Su, C., Equialet coditios of comlete coergece for ideedet weighted sums. Sciece Chia ( series A, 1998; 41:

21 Kamo Budsaba 161 [4] Stei, C., A boud o the error i the ormal aroximatio to the distributio of a sum of deedet radom ariables. I Proceedigs of the Sixth Bereley Symosium o Mathematical Statistics ad Probability, Bereley: Uersity of Califora Press.,197; : [5] Bradley, R. C., O the sectral desity ad asymtotic ormality of wealy deedet radom fields. J. Theor. Probab., 199; 5: [6] Bradley, R. C., Equialet mixig coditios for radom fields. A. Probab., 1993; 1: [7] Bryc, W., Smolesi W., Momet coditios for almost sure coergece of wealy correlated radom ariables. Proc. Amer. Math. Soc., 1993; 19: [8] Shachao, Y., Some momet iequalities for artial sums of radom ariables ad their alicatios. Chiese Sci. Bull.,1998; 43: [9] Gut, A., Peligrad, M., Almost-sure results for a class of deedet radom ariables. J. Theoret. Probab., 1999; 1: [10] Wu, Q., Some coergece roerties for ~ -mixig radom ariables sequeces. J. Egg. Math. (Chiese, 001; 18: [11] Wu, Q., Coergece for weighted sums of ~ -mixig radom sequeces. Math. Al. (Chiese, 00; 15: 1-4. [1] Shixi, G., Almost sure coergece for ~ -mixig radom ariable sequeces. Statist. Probab. Lett., 004; 67: [13] Ute, S. Peligrad, M., Maximal iequalities ad a iariace ricile for a class of wealy deedet radom ariables. J. Theoret. Probab., 003; 16: [14] Dehua, Q. ad Shixi, G., Coergece for arrays of ~ -mixig radom ariables. Acta Mathematica Scietia. (Chiese, 005; 5A(1: [15] Dehua, Q. ad Shixi, G., Coergece for weighted sums of ~ -mixig radom ariables sequeces. J. of Math. (PRC, 008; 8: [16] Dehua, Q. ad Shixi, G., The Háec-Rèyi iequqlity ad strog law of large umbers for ~ -mixig sequeces. Joural of Mathematics i Practice ad Theory (Chiese, 007; 37: [17] Seeta E., Regularly aryig fuctio, Lecture Notes i Math. 508, Sriger, Berli, 1976.

22 16 Thailad Statisticia, 01; 10(1: [18] Bai Z.D., Su C., The comlete coergece for artial sums of i.i.d. radom ariables. Sci Sica(Ser A, 1985; 8: [19] Li D., Rao M. B., Jiag T., Wag., Comlete coergece ad almost sure coergece of weighted s ums of radom ariables. J. Theoret. Probab., 1995; 8: