Research Article Complete Convergence for Maximal Sums of Negatively Associated Random Variables

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1 Hidawi Publishig Corporatio Joural of Probability ad Statistics Volume 010, Article ID , 17 pages doi: /010/ Research Article Complete Covergece for Maximal Sums of Negatively Associated Radom Variables Victor M. Kruglov Departmet of Statistics, Faculty of Computatioal Mathematics ad Cyberetics, Moscow State Uiversity, Vorobyovy Gory, GSP-1, 11999, Moscow, Russia Correspodece should be addressed to Victor M. Kruglov, Received 4 December 009; Accepted 1 April 010 Academic Editor: Mohammad Fraiwa Al-Saleh Copyright q 010 Victor M. Kruglov. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited. Necessary ad sufficiet coditios are give for the complete covergece of maximal sums of idetically distributed egatively associated radom variables. The coditios are expressed i terms of itegrability of radom variables. Proofs are based o ew maximal iequalities for sums of bouded egatively associated radom variables. 1. Itroductio The paper by Hsu ad Robbis 1 iitiated a great iterest to the complete covergece of sums of idepedet radom variables. Their research was cotiued by Erdös, 3, Spitzer 4, ad Baum ad Katz 5. Kruglov et al. 6 proved two geeral theorems that provide sufficiet coditios for the complete covergece for sums of arrays of row-wise idepedet radom variables. I the paper of Kruglov ad Volodi 7, a criterio was proved for the complete covergece of sums of idepedet idetically distributed radom variable i a rather geeral settig. Taylor et al. 8 ad Che et al. 9, 10 demostrated that may ow sufficiet coditios for complete covergece of sums of idepedet radom variables ca be trasformed to sufficiet coditios for the complete covergece of sums of egatively associated radom variables. Here we give ecessary ad sufficiet coditios for the complete covergece of maximal sums of egatively associated idetically distributed radom variables. They resemble the criterios preseted by Baum ad Katz 5 ad by Kruglov ad Volodi 7 for the complete covergece of sums of idepedet idetically distributed radom variables. Theorems.3 ad.5 are our mai results. Theorem.3 is ew eve for idepedet radom variables.

2 Joural of Probability ad Statistics I what follows we assume that all radom variables uder cosideratio are defied o a probability space Ω, F,P. We use stadard otatios, i particular, I A deotes the idicator fuctio of a set A Ω. Recall the otio of egatively associated radom variables ad some properties of such radom variables. Defiitio 1.1. Radom variables X 1,...,X are called egatively associated if Cov f X i1,...,x i,g X j1,...,x jm for ay pair of oempty disjoit subsets A i 1,...,i ad B j 1,...,j m, m, of the set 1,..., ad for ay bouded coordiate-wise icreasig real fuctios f x i1,...,x i ad g x j1,...,x jm,x 1,...,x R,. Radom variables X, N 1,,..., are egatively associated if for ay N radom variables X 1,...,X are egatively associated. I this defiitio the coordiate-wise icreasig fuctios f ad g may be replaced by coordiate-wise decreasig fuctios. Ideed, if f ad g are coordiate-wise decreasig fuctios, the f ad g are coordiate-wise icreasig fuctios ad the covariace 1.1 coicides with the covariace for f ad g. Theorem A. Let X, N, be egatively associated radom variables. The for every a,b R,a b, the radom variables Y a I,a X X I a,b X b I b, X, N, are egatively associated. For every N ad x 1,...,x R, the iequalities PX 1 x 1,...,X x PX 1 x 1,...,X x PX x, PX x 1. hold. Proof. It ca be foud i Taylor et al. 8. Theorem B. Let X be egatively associated radom variables. Let X, N, be idepedet radom variables such that X ad X are idetically distributed for every 1,...,.The E expx 1 X E exp X 1 X. 1.3 If E X p < ad EX 0 for all 1,...,ad for some p 1, the r p E max X E 1 r X p, N. 1.4 Proof. It ca be foud i Qi-Ma Shao 11.

3 Joural of Probability ad Statistics 3. Mai Results Our basic theorems will be stated i terms of special fuctios. They were itroduced i Kruglov ad Volodi 7. Defiitio.1. A oegative fuctio h x,x 0,, belogs to the class H q for some q 0, if it is odecreasig, is ot equal to zero idetically, ad y /q 1 lim sup y h y h x lim sup x h x <, y h x x /q dx <..1. The class H q cotais all odecreasig oegative fuctios slowly varyig at ifiity which are ot equal to zero idetically, ad i particular, l β 1 x with β>0. The fuctios x α ad x α l β 1 x with α 0, /q 1 ad β>0arealsoih q. Remar.. If a oegative fuctio h x,x 0,, is odecreasig ad satisfies coditio.1, the h x Δx l.3 for all x greater tha some x 0 1 ad for some Δ > 0adl>0. Proof. We may assume that h x > 0 for all x greater tha some x 0 1. From coditio.1, it follows that sup x x0 h x /h x d<. Choose a umber l>0 such that d l. If x x 0, the x 0 x< x 0 for some N ad h x h x 0 d h x 0 l h x 0 l x l h x 0. Iequality.3 holds for all x x 0 with Δ l h x 0. Theorem.3. Let X, N, be egatively associated idetically distributed radom variables, S X 1 X, 0 <q<,r > 1. Let h x,x 0,, be a fuctio which is odecreasig, is ot equal to zero idetically, ad satisfies coditio.1. The the followig coditios are equivalet: E X 1 rq h X 1 q <, EX 1 0, for q 1,.4 r h P max S >ε 1/q <, ε >0,.5 1 r h P sup 1/q S >ε <, ε >0..6

4 4 Joural of Probability ad Statistics Corollary.4. Let X, N, be egatively associated idetically distributed radom variables, S X 1 X, 0 <q<,r >1,β 0. The followig coditios are equivalet: E X 1 rq l X 1 βq <, EX 1 0, for q 1, r l β P max S >ε 1/q <, ε >0, 1 r l β P sup 1/q S >ε <, ε >0..7 ApartofTheorem.3 ca be geeralized to a larger rage of r, r 1, uder additioal restrictios o fuctios h x,x 0,. Theorem.5. Let X, N, be egatively associated idetically distributed radom variables, S X 1 X,h H q, 0 <q<,r 1. The the followig coditios are equivalet: E X 1 rq h X 1 q <, EX 1 0, for q 1,.8 r h P max S >ε 1/q <, ε > Corollary.6. Let X, N, be egatively associated idetically distributed radom variables, S X 1 X, 0 <q<,β 0. The followig coditios are equivalet: E X 1 q l X 1 βq <, EX 1 0, for q 1, l β P max S >ε 1/q <, ε > Proof of Theorem.3. The theorem is obvious if the radom variable X 1 is degeerate, that is, PX 1 cost. 1. From ow o we suppose that the radom variable X 1 is ot degeerate. Deote r 1 r l r 1 δ 4 r l q 4q r l if 0 <rq<1, if 1 rq, if rq >,.11

5 Joural of Probability ad Statistics 5 where l is the same as i.3. Defie the fuctio f x,x R, ε 1/q δ if x>ε 1/q δ, f x x if x ε 1/q δ, ε 1/q δ if x< ε 1/q δ,.1 where ε>0 is a fixed umber. By Theorem A the radom variables Y f X, 1,...,, are egatively associated. Put S Y 1 Y, 1,...,.Note that Y 1 X 1 ad Y 1 X 1 as. If E X 1 rq <, the by the domiated covergece theorem we have lim E Y 1 rq E X 1 rq, lim E Y 1 EY 1 rq E X 1 EX 1 rq if rq Prove that.4.5. Assume that 0 <rq<1. The probability Pmax 1 S >ε 1/q ca be estimated as follows: P max S >ε 1/q 1 P max S >ε 1/q 1 X Y P max S >ε 1/q 1 P max S >ε 1/q 1 X / Y P max X >ε 1/q δ We ited to use Lemma 3.1 from the third part of the paper. Put γ rq,x ε 1/q,c ε 1/q r 1 / r l. From.13 it follows that the iequality E Y 1 γ < E X 1 γ holds for all N greater tha some 0. By iequality 3.1, weget,forall> 0, P max S >ε 1/q x exp 1 c x xc γ 1 c l E Y 1 γ 1 r l exp r 1 r l ε rq r 1 l r 1 rq 1 E X 1 rq r l rq 1 r 1 1 r l ε rq r 1 rq 1 r l / r 1 exp r l C r l. r 1 E X 1 rq r l rq 1.15 These iequalities ad.3 imply that r h P max S >ε 1/q ΔC r l r l <..16

6 6 Joural of Probability ad Statistics Sice Pmax 1 X >ε 1/q δ P X 1 >ε 1/q δ, we obtai 0 1 r h P max X >ε 1/q δ 1 r 1 h P X 1 >ε 1/q δ 0 1 X 1 E εδ rq X 1 q h εδ..17 The last iequality holds by Lemma 3..From.1 it follows that X 1 E εδ rq X 1 q h εδ cost. E X 1 rq h X 1 q..18 Coditio.4 ad iequalities imply.5 for 0 <rq<1. Now assume that rq 1. First we cosider the case 1 rq. Note that P max S >ε 1/q P max S ES >ε 1/q max ES By Lemma 3.4, we have max 1 E S 1/q 0as, ad hece P max S >ε 1/q P maxs ES > ε1/q for all N greater tha some 0. The probability o the right-had side ca be estimated as follows: P max S ES ε 1/q > P max S 1 1 ES ε 1/q > X Y P max S ES ε 1/q > X / Y 1 P max S 1 ES ε 1/q > P max X >ε 1/q δ. 1.1

7 Joural of Probability ad Statistics 7 I order to apply Lemma 3.1, putγ rq,x ε 1/q /,c ε 1/q r 1 / 4 r l. From.13 it follows that the iequality E Y 1 EY 1 γ < E X 1 EX 1 γ holds for all N greater tha some 0 0. By iequality 3. we get, for > 0, P maxs 1 ES > ε1/q x 4 exp γ 1 c x c l xc γ 1 E Y 1 EY 1 γ 1 4 exp r l r 1 r l r 1 l ε rq r 1 rq 1 E X 1 EX 1 rq 4 r l rq 1 r 1 1 r l ε rq r 1 rq 1 r l / r 1 4 exp r l C r 1 E X 1 EX 1 rq 4 r l rq 1 1 r l.. These iequalities ad.3 imply r h P max S ES ε 1/q > ΔC 1 r l r l < From this iequality ad.17 with 0 istead of 0,.0 ad.1, it follows that.5 holds for 1 rq. The case rq > ca be cosidered i the same way. I order to apply Lemma 3.1, putγ,x ε 1/q /,c ε 1/q q / 4q r l. From.13 it follows that the iequality E Y 1 EY 1 < E X 1 EX 1 holds for all N greater tha some 0 0. By iequality 3., weget,for> 0, P maxs 1 ES > ε1/q x 4 exp c x c l xc E Y 1 EY 1 1 q r l q r l 4 exp q q l ε q 16E X 1 EX 1 q r l /q 1 1 q r l ε q r l / q q 4 exp r l C q 16E X 1 EX 1 r l. q r l.4 These iequalities ad.3 imply r h P max S ES ε 1/q > ΔC r l r l <

8 8 Joural of Probability ad Statistics From this iequality ad.17 with 0 istead of 0,.0 ad.1, it follows that.5 holds for rq >. Prove that.5.6. Note that r h P sup 1/q S >ε 3 j 1 j 1 j 1 r h P sup 1/q S >ε j 1 r 1 h j 1 P sup 1/q S >ε j 1 j j 1 r 1 h j 1 P max 1/q S >ε i i 1 j 1 i 1 j 1 i j i j 1 r 1 h j 1 P r 1 i 1 r 1 h i 1 P r 1 1 i 1 max i i 1 1/q S >ε max S >ε i/q i i 1..6 The last series ca be estimated as follows: i 1 r 1 h i 1 P max S >ε i/q i i 1 i 1 5r 3 h 5 max 1, r 5r 3 h 5 max 1, r i i 1 i 1 1 r h P r h P max S >ε /q 1/q 1 max S >ε /q 1/q. 1.7 Coditio.5 ad these iequalities imply.6. Prove that.6.4. The sequece Psup 1/q S >ε 1 decreases to zero for ay ε>0. Ideed, if the sequece coverges to a umber a>0, the a r h r h P sup 1/q S >ε <..8 Note that Pmax S >ε 1/q Psup 1/q S >ε 0as.

9 Joural of Probability ad Statistics 9 With the help of 1., weobtai P max X > 1/q 1 P max X 1/q < < 1 P max X > 1/q 1 P < 1 P X 1/q 1 1 P X 1 > 1/q, max X 1/q < P X 1/q 1 1 P X 1 > 1/q..9 Deote a PX 1 > 1/q ad a P X 1 > 1/q. Note that P X 1 > 1/q a a ad 1 1 a ± Pmax < X > 1/q Pmax S > 1/q /. Sice Pmax S > 1/q / 0ada ± P X 1 > 1/q 0as, the l a ± > l ad 0 a a < 1 for all N greater tha some 0. By the iequalities l x x for x 0, 1 ad 1 e x e x x for x 0, we obtai a ± l 1 a ± 1 e l a± 1 1 a ± P max X > 1/q 1.30 for all > 0,ad 0 1 r 1 h P X 1 > 1/q r 1 h a a r h P max X > 1/q < 4 r h P max S > 1/q 4 r h P max 1/q 1/q S > 4 r h P sup 1/q 1/q S > <..31 It follows by Lemma 3.3 that E X 1 rq h X 1 q <. Now we will prove that a EX 1 0 provided that 1 q<. Assume that a / 0. Sice a S a S, the 1 P S a S > a 1/q P S a > a 1/q P S > a 1/q,.3 4 4

10 10 Joural of Probability ad Statistics ad hece r h r h P S a > a 1/q r h P S > a 1/q But this cotradicts the covergece of the series o the right-had side of the iequality. The equality EX 1 0 is proved. Proof of Theorem.5. Both theorems overlap. We eed to cosider oly the case r 1. Prove that.8.9. Defie radom variables Z ε 1/q I, ε 1/q X X I ε 1/q,ε 1/q X ε 1/q I ε 1/q, X, N. By Theorem A the radom variables Z, N, as well as Z EZ, N, are egatively associated. Deote S Z 1 EZ 1 Z EZ. By Theorem B there exist idepedet radom variables X 1,...,X such that the radom variables Z EZ ad X are idetically distributed for all 1,...,, ad E max 1 S E X 1 X. Similarly to.1, we ca prove that P max S >ε 1/q P max S 1 1 ES ε 1/q > P max X >ε 1/q 1.34 for all N grater tha some 0. I the same way as.17, oe ca prove that 0 1 h P max X >ε 1/q h P X 1 >ε 1/q X 1 q E X 1 q ε h ε <..35 With the help of the Marov iequality, we obtai P maxs 1 ES > ε1/q 4ε /q E maxs ES 1 8ε /q E X 1 X.36 8ε 1 /q E Z 1 ZY 1. From.34 ad.35, it follows that.9 holds if the series h /q E Z 1 EZ 1 coverges. This series ca be estimated as follows: h E Z /q 1 EZ 1 h E Z 1 /q h E X /q 1 I X 1 ε 1/q ε h P X 1 >ε 1/q..37

11 Joural of Probability ad Statistics 11 Rewrite the first summad o the right-had side i the followig way: h E X /q 1 h I X1 ε 1/q E X /q 1 I ε 1 1/q < X 1 ε 1/q h E X /q 1 I ε 1 1/q < X 1 ε 1/q..38 From.1 ad., it follows that there exist umbers 0 N, 0 >, ad C>0 such that h Ch, h x x /q dx Ch 1 /q, For ay N, we have h /q /q h x dx /q x/q h x dx <. x.40 /q If > 0, the h /q /q h x x /q dx /q Ch 1 /q /q C h /q..41 With the help of these estimates, we get h E X /q 1 0 I X1 ε 1/q /q /q C 0 1 h x dx x/q h 1 1 E X /q 1 1 I ε 1 1/q < X 1 ε 1/q..4 The last series ca be estimated as follows: 0 1 h 1 1 E X /q 1 1 I ε 1 1/q < X 1 ε 1/q ε E /q 1 ε 1 X q ε h 1 q X ε /q 1 ε E 1 q ε X 1 h 1 q X 1 <. I ε 1 1/q < X 1 ε 1/q.43 As a result we get that h /q E X 1 I X1 ε 1/q <. Taig accout of.35 ad.37, weseethat h /q E Z 1 EZ 1 <.

12 1 Joural of Probability ad Statistics Prove that.9.8 for r 1. Note that h P max X > /q h P max S > /q < 1 h j h P j P max j S > /q j 1/q j max S > /q 1/q 1 <..44 Deote b PX 1 > /q ad b P X 1 > /q. Note that P X 1 > /q b b. With the help of 1., oe ca prove the iequality 1 1 b ± Pmax < X > /q. Sice Pmax < X > /q 0adb ± P X 1 > /q 0as, the l b ± > l ad 0 b b < 1 for all N greater tha some 0. By the iequalities l x x for x 0, 1 ad 1 e x e x x for x 0, we obtai h P X 1 > /q b ± l 1 b ± 1 e l b ± 1 1 b ± P max X > /q < h b b, 4 h P max X > /q <. <.45 By Lemma 3.3, we have that E X 1 q h X 1 q <. I the same way as i the proof of the previous theorem, oe ca prove that EX 1 0ifq Auxiliary Results Let X 1,...,X be radom variables. Deote S X 1 X for 1,..., ad A,γ E X 1 γ E X γ,b,γ E X 1 EX 1 γ E X EX γ for some γ 0,, cosh x e x e x /,x R,.

13 Joural of Probability ad Statistics 13 Lemma 3.1. If egatively associated radom variables X 1,...,X are bouded by a costat c>0, the x P max S >x exp 1 c x xc γ 1 c l 1, 0 <γ 1, 3.1 A,γ x P max S ES >x 4 exp 1 γ 1 c x xc γ 1 c l 1, 1 γ, 3. B,γ for ay umber x>0. Proof. Prove Iequality 3.1. It is easily verified that max 1 S >x X >x X >x where X max0,x ad X max0, X. With the help of Marov iequality, we get P max S >x P X >x 1 e hx E exp h X P X >x e hx E exp h X 3.3 for ay h>0. By Theorem A, radom variables X, 1,...,, as well as X, 1,...,, are egatively associated. By Theorem B, the iequality E exp h X ± 1 X X± E exp ± h 1 X ± 3.4 holds where radom variables X 1,...,X are idepedet, ad for ay 1,...,,radom variables X ad X are idetically distributed. It follows that P max S >x e hx 1 e hx Ee h X e hx Ee hx e hx Ee h X Ee hx e hx Ee h X. 3.5

14 14 Joural of Probability ad Statistics Further we ca proceed as i Prohorov 1, Fu ad Nagaev 13, ad Kruglov 14. Assume that 0 < γ 1. The fuctio e hx 1 /x γ, 0 x c, icreases ad hece e hx 1 /x γ e hc 1 /c γ. With the help of this iequality, we obtai Ee h X E 1 X γ e h X 1 X γ 1 c γ e hc 1 E X γ exp c γ e hc 1 A,γ. 3.6 From this iequality ad from 3.5, it follows P max S >x exp hx c γ e hc 1 A,γ Put h c 1 l xc γ 1 /A,γ 1. As a result we obtai 3.1. Now we assume that 1 γ. By the Marov iequality we get P max S ES >x E cosh h max 1 S ES 1 cosh hx 3.8 for ay h>0. By Theorem B, the iequality r E max S ES E X EX X EX r 3.9 holds for ay r>1. Deote S X 1 X. Note that cosh x cosh x for ay x R. With these remars, we obtai E cosh h max S ES 1 1 r 1 E h max S ES r 1 r! E h S ES 1 r r! r 1 E cosh h S ES E cosh h S ES, 3.10 ad hece P max S ES >x E cosh h S ES 1 cosh hx e hx Ee h S ES Ee h S ES. 3.11

15 Joural of Probability ad Statistics 15 By the iequalities α e α 1 ad e α 1 α cosh α 1forα R, we obtai Ee h S ES Ee h X EX exp E e h X EX 1 exp E e h X EX 1 h X EX exp E cosh h X EX Put f α cosh α 1 α γ for α / 0adf 0 1/ ifγ, ad f 0 0if1 γ<. It ca be easily verified that the fuctio f is cotiuous, eve, ad icreases o 0,. Note that X EX cfor all 1,...,.With these remars, we obtai cosh h X EX 1 cosh h X EX 1 h X EX γ h X EX γ cosh hc 1 c γ E X EX γ, 3.13 ad hece Ee h S ES exp cosh hc 1 c γ B,γ I the same way, oe ca prove the iequality Ee h S ES exp cosh hc 1 c γ B,γ From these iequalities ad from 3.11, it follows that P max S ES >x 4 exp hx cosh hc 1 c γ B,γ 1 4 exp hx e hc 1 c γ B,γ Put h c 1 l xc γ 1 /B,γ 1. As a result we obtai 3.. Lemma 3.. Let h x,x 0,, be a odecreasig oegative fuctio, ξ be a oegative radom variable, r 1, ad q > 0. The r 1 h P ξ> 1/q E ξ rq h ξ q Proof. It ca be foud i Kruglov ad Volodi 7.

16 16 Joural of Probability ad Statistics Lemma 3.3. Let h x,x 0,, be a odecreasig oegative fuctio possessig property.1, ξ be a oegative radom variable, r 1,q > 0, ad b 1 a ubouded odecreasig sequece of positive umbers such that b bb for all N ad for some umber b>0,b 0 0. The there exist umbers 0 N ad d>0 such that de ξ rq h ξ q I ξ b0 b r 1 h b b b 1 P ξ>b 1/q Proof. It ca be foud i Kruglov ad Volodi 7. The ext lemma was proved i Kruglov ad Volodi 7 uder a additioal restrictio. Lemma 3.4. Let X, N, be idetically distributed radom variables such that E X 1 q < for some q 0, ad EX 1 0 if 1 q <. Defie the fuctio f x ε 1/q I, ε 1/q x xi ε 1/q,ε 1/q x ε 1/q I ε 1/q, x,x R, where ε>0is a fixed umber. The 1 lim max 1/q 1 j 1 Ef X j Proof. Note that max Ef X j 1 Ef X1 E X1 I X1 ε 1/q ε P 1/q X 1 >ε 1/q, j 1 1 lim 1/q 1/q 1 P X 1 >ε 1/q lim P X 1 q >ε q It suffices to prove that lim E X 1/q 1 I X1 ε 1/q Suppose that 0 <q<1. For ay δ>0, there exist 0 N such that E X 1 q I X1 >ε 1/q <δ.we get, for ay 0, E X 1/q 1 I X1 ε 1/q E X 1/q 1 I X1 ε 1/q ε 1 q E X 0 1 q I 1/q ε 0 < X 1, ε 1/q 3. ad hece lim sup E X 1/q 1 I X1 1/q ε 1 q δ. 3.3 This implies 3.1,siceδ>0 ca be chose arbitrarily small.

17 Joural of Probability ad Statistics 17 If 1 q<adex 1 0, the we get E X1 I 1/q X1 ε 1/q E X1 I 1/q X1 >ε 1/q ε 1 q E X 1 q I X1 >ε 1/q Acowledgmets The author is grateful to a aoymous referee for careful readig of the paper ad useful remars. This research was supported by the Russia Foudatio for Basic Research, projects a, a, Uzbeista. Refereces 1 P. L. Hsu ad H. Robbis, Complete covergece ad the law of large umbers, Proceedigs of the Natioal Academy of Scieces of the Uited States of America, vol. 33, pp. 5 31, P. Erdös, O a theorem of Hsu ad Robbis, Aals of Mathematical Statistics, vol. 0, pp , P. Erdös, Remar o my paper O a theorem of Hsu ad Robbis, Aals of Mathematical Statistics, vol. 1, article 138, F. Spitzer, A combiatorial lemma ad its applicatio to probability theory, Trasactios of the America Mathematical Society, vol. 8, pp , L. E. Baum ad M. Katz, Covergece rates i the law of large umbers, Trasactios of the America Mathematical Society, vol. 10, pp , V. M. Kruglov, A. I. Volodi, ad T.-C. Hu, O complete covergece for arrays, Statistics ad Probability Letters, vol. 76, o. 15, pp , V. M. Kruglov ad A. I. Volodi, Covergece rates i the law of large umbers for arrays, Probability ad Mathematical Statistics, vol. 6, o. 1, pp , R. L. Taylor, R. F. Patterso, ad A. Bozorgia, A strog law of large umbers for arrays of rowwise egatively depedet radom variables, Stochastic Aalysis ad Applicatios, vol. 0, o. 3, pp , P. Che, N.-C. Yu, X. Liu, ad A. Volodi, O complete covergece for arrays of rowwise egatively associated radom variables, Theory of Probability ad Its Applicatio, vol. 5, pp , P. Che, N.-C. Yu, X. Liu, ad A. Volodi, O complete covergece for arrays of rowwise egatively associated radom variables, Probability ad Mathematical Statistics, vol. 6, pp , Q.-M. Shao, A compariso theorem o momet iequalities betwee egatively associated ad idepedet radom variables, Joural of Theoretical Probability, vol. 13, o., pp , Yu. V. Prohorov, Some remars o the strog law of large umbers, Theory of Probability ad Its Applicatios, vol. 4, pp. 15 0, D. H. Fu ad S. V. Nagaev, Probabilistic iequalities for sums of idepedet radom variables, Theory of Probability ad Its Applicatios, vol. 16, pp , V. M. Kruglov, Stregtheig the Prohorov arcsie iequality, Theory of Probability ad Its Applicatios, vol. 50, o. 4, pp , 005.