Research Article On the Strong Convergence and Complete Convergence for Pairwise NQD Random Variables

Transcription

1 Abstract ad Applied Aalysis Volume 204, Article ID , 7 pages Research Article O the Strog Covergece ad Complete Covergece for Pairwise NQD Radom Variables Aitig She, Yig Zhag, ad Adrei Volodi 2 School of Mathematical Sciece, Ahui Uiversity, Hefei 23060, Chia 2 Departmet of Mathematics ad Statistics, Uiversity of Regia, Regia, SK, Caada S4S 0A2 Correspodece should be addressed to Aitig She; sheaitig4@26.com Received 5 December 203; Accepted 7 April 204; Published 8 May 204 Academic Editor: Agelo Favii Copyright 204 Aitig She et al. 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 work is properly cited. Let {a, }be a sequece of positive costats with a / ad let {X, X, }be a sequece of pairwise egatively quadrat depedet radom variables. The complete covergece for pairwise egatively quadrat depedet radom variables is studied uder mild coditio. I additio, the strog laws of large umbers for idetically distributed pairwise egatively quadrat depedet radom variables are established, which are equivalet to the mild coditio P( X >a ) <. Our results obtaied i the paper geeralize the correspodig oes for pairwise idepedet ad idetically distributed radom variables.. Itroductio Throughout the paper, let {a, }be a sequece of positive costats with a /,adlet{x, X, }be a sequece of pairwise i.i.d. radom variables. Deote S = i= X i for each.now,wecosiderthefollowigassumptios: (i) P( X > a )<; (ii) S /a 0a.s.; (iii) i= X i /a 0a.s. Recetly, Sug [] proved that the three assumptios above are equivalet for pairwise i.i.d. radom variables. I additio, he preseted some results o complete covergece for pairwise i.i.d. radom variables. For more details about the strog law of large umbers ad complete covergece for idepedet radom variables or depedet radom variables, oe ca refer to Etemadi [2], Wag et al. [3], Che et al. [4], Tag [5],adsoforth. We poit out that the keys to the proofs of the mai results of Sug [] are the Khitchie-Kolmogorov-type covergece theorem ad the secod Borel-Catelli lemma for pairwise idepedet evets (e.g., see Theorem i [6] or Theorem i [7]), while these are ot proved for pairwise egatively quadrat depedet radom variables (pairwisenqd,ishort;seedefiitio ). If we wat to geeralizethemairesultsofsug[] to the case of pairwise NQD radom variables, we should propose ew methods or prove the Khitchie-Kolmogorov-type covergece theorem ad the secod Borel-Catelli lemma for pairwise NQD radom variables. The aswer is positive. Firstly, let us recall the cocept of pairwise egatively quadrat depedet radom variables as follows. Defiitio. The pair (X, Y) of radom variables X ad Y is said to be egatively quadrat depedet (NQD, i short), if, for all x, y R, P(Xx,Yy)P(X x) P(Yy). () A sequece of radom variables {X, } is said to be pairwise NQD, if (X i,x j ) is NQD for every i =j, i, j =, 2,... A array {X i,i, }of radom variables is called rowwise pairwise NQD radom variables if for every, {X i,i }is a sequece of pairwise NQD radom variables. The cocept of pairwise NQD radom variables was itroduced by Lehma [8], which icludes pairwise idepedet radom sequece ad some egatively depedet

2 2 Abstract ad Applied Aalysis sequeces, such as egatively associated sequeces (see [9 3]), egatively orthat depedet sequeces (see [9, 4 8]), ad liearly egative quadrat depedet sequeces (see [9 2]). Hece, studyig the probability limitig behavior of pairwise NQD radom variables ad its applicatios i probability theory ad mathematical statistics are of great iterest. May authors have dedicated themselves to the study of it. Matula [0] gaied the Kolmogorov-type strog law of large umbers for the idetically distributed pairwise NQD sequeces; Wu [22] gavethegeeralizedthreeseries theorem for pairwise NQD sequeces ad proved the Marcikiewicz strog law of large umbers; Che [23] discussed Kolmogorov-Chug strog law of large umbers for the oidetically distributed pairwise NQD sequeces uder very mild coditios; Wa [24] ad Huag et al. [25] obtaied the complete covergece for pairwise NQD radom sequeces; Wag et al. [26], Li ad Yag [27], Ga ad Che [28], Shi [29], Xu ad Tag [30], ad Tag [3] studied the strog covergece properties for pairwise NQD radom variables; Sug [2] establishedthel r covergece for weighted sums of arrays of rowwise pairwise NQD radom variables uder weaker uiformly itegrable coditios;adsoo.themaipurposeofthepaperisto establish the secod Borel-Catelli lemma for pairwise NQD radom variables ad geeralize the mai results of Sug [] tothecaseofpairwisenqdradomvariableswithoutaddig ay extra coditios. Our mai results are as follows. The first two results are thecompletecovergeceforpairwisenqdradomvariables. Theorem 2. Let {a, }be a sequece of positive costats with a /.Let{X, X, }be a sequece of pairwise NQD radom variables with idetical distributio. If P( X > a )<,the P( [X i EX i I( X i a )] >a ε)< ε> 0. Theorem 3. Let {a, }be a sequece of positive costats with a /. Let{X, X, } be a sequece of pairwise NQD radom variables with idetical distributio. If P( X > a )<,the (2) P(max S k >a ε)< ε >0. (3) k The followig two theorems are the results o strog covergece for pairwise NQD radom variables. Theorem 4. Let {a, }be a sequece of positive costats with a /. Let{X, X, } be a sequece of pairwise NQD radom variables with idetical distributio. The, the followig statemets are equivalet: (i) P( X > a )<, (ii) (/a ) i= [X i EX i I( X i a )] 0 a.s. Theorem 5. Let {a, }be a sequece of positive costats with a /. Let{X, X, }be a sequece of pairwise NQD radom variables with idetical distributio. The, the followig statemets are equivalet: (i) P( X > a )<, (ii) S /a 0a.s., (iii) i= X i /a 0a.s. With Theorem 5 ad the secod Borel-Catelli lemma for pairwise NQD radom variables (see Corollary 6) i had, we ca get the followig result for pairwise NQD radom variables. Corollary 6. Let {a, }be a sequece of positive costats with a /.Let{X, X, }be a sequece of pairwise NQD radom variables with idetical distributio ad E X =. The, lim a i= X i = 0 a.s. iff P( X >a )<; lim sup S = a.s. iff a P( X >a )=. Remark 7. Theorems 2 ad 3 deal with the complete covergece for pairwise NQD radom variables. Theorems 4 ad 5 deal with the strog laws of large umbers for pairwise NQD radom variables, which are equivalet to the mild coditio P( X > a ) <. Pairwise NQD is a very wide depedece structure, which icludes idepedet sequece as a special case. Hece, Theorems 2 5 geeralize the correspodig oes for pairwise i.i.d. radom variables tothecaseofpairwisenqdradomvariables. Remark 8. Uder the coditios of Theorem 3 ad a 2 Ca, we ca get the Marcikiewicz-Zygmud-type strog law of largeumbersforpairwisenqdradomvariablesasfollows: a i= (4) X i 0 as. (5) Remark 9. For a sequece {X, X, } of pairwise i.i.d. radom variables with E X <, Etemadi[2] proved that i= (X i EX i )/ 0 a.s. Note that E X < is equivalet to P( X > ) < ad E X < implies i= EX ii( X i >i)/ 0. Hece, Etemadi s strog law of large umbers follows from Theorem 4 with a =. Remark 0. Note that lim sup ( S /a ) = a.s. is equivalet to P( S >αa, i.o.) =for ay α>0.hece, Corollary 6 improves the correspodig result of Kruglov [32]. Throughout the paper, let I(A) be the idicator fuctio of the set A. C deotes a positive costat ot depedig o, which may be differet i various places. Deote a 0 =0, x + =xi(x 0),adx = xi (x < 0).

3 Abstract ad Applied Aalysis 3 2. Prelimiaries I this sectio, we will preset some importat lemmas which will be used to prove the mai results of the paper. ThefirstthreelemmascomefromSug[]. Lemma (cf.[]). Let {a, } be a sequece of positive costats with a /. The the followig properties hold. (i) {a, }isastrictly icreasig sequece with a a. (ii) P(X>a )<if ad oly if P(X>2a )<. (iii) P(X>a )<if ad oly if P(X>αa )< for ay α>0. Lemma 2 (cf. []). If {a, } is a sequece of positive costats with a / ad X is a radom variable, the a E X I ( X a ) =0 P ( X >a ). (6) Lemma 3 (cf. []). Let {a, } be a sequece of positive costats with a / ad X is a radom variable. If P( X > a )<,the(/a )E X I( X a ) 0. The ext oe is the basic property for pairwise NQD radom variables, which was give by Lehma [8] as follows. Lemma 4 (cf. [8]). Let X ad Y be NQD; the (i) EXY EXEY; (ii) P(X > x, Y > y) P(X > x)p(y > y), foray x, y R; (iii) if f ad g are both odecreasig (or oicreasig) fuctios, the f(x) ad g(y) are NQD. The followig oe is the geeralized Borel-Catelli lemma, which was obtaied by Matula [0]. Lemma 5 (cf. [0]). Let {A, }be a sequece of evets. (i) If P(A )<,thep(a, i.o.) = 0. (ii) If P(A k A m ) P(A k )P(A m ) for k =m ad P(A )=,thep(a, i.o.) =. With the geeralized Borel-Catelli lemma accouted for, we ca establish the secod Borel-Catelli lemma for pairwise NQD radom variables as follows. Corollary 6 (secod Borel-Catelli lemma for pairwise NQD radom variables). Let {a, } be a sequece of positive costats with a /.Let{X, }be a sequece of pairwise NQD radom variables. The X a 0 a.s. P ( X >a ) <. (7) Proof.. By Lemma, P( X >a )<is equivalet to P( X >a ε)<for all ε>0,whichyields that X /a 0a.s. by Borel-Catelli lemma.. LetX /a 0 a.s., which implies that X + /a 0 a.s. ad X /a 0a.s. For ay ε>0, deote Hece, A () ={ X+ > ε a 2 }, A (2) ={ X > ε }. (8) a 2 P {A (j), i.o.} =0, j=,2. (9) By Lemma 4(iii), we ca see that {X +, }ad {X, } are both sequeces of pairwise NQD radom variables. It follows by Lemma 4(ii) that, for ay k =m, P(A k (j) A m (j)) P (A k (j)) P (A m (j)), j =, 2. (0) By Lemma 5(ii) ad (9)-(0), we ca see that P(A (j)) < for j=,2.hece, P( X >a ε) P(A ())+ P(A (2)) < for ay ε>0, () which is equivalet to P( X >a )<by Lemma. This completes the proof of the corollary. The last oe is the Kolmogorov-type strog law of large umbers for pairwise NQD radom variables obtaied by Che [23], which plays a importat role i provig the mai results of the paper. Lemma 7 (cf. [23]). Let {X, }be a sequece of pairwise NQD radom variables with Var(X )<for each.let {a, }be a sequece of real umbers satisfyig 0<a.Supposethat (i) sup a i= E X i EX i <; (ii) Var(X )/a 2 <. The a i= (X i EX i ) 0a.s. 3. Proofs of Theorems 2 5 Proof of Theorem 2. Note that the coditio a / implies that =i a 2 =i i 2 i2 ai 2 2 ai 2 =i 2 i2 a 2 i For fixed, deote for ithat 2 i = 2i ai 2. (2) Y i = a I(X i < a )+X i I( X i a )+a I(X i >a ). (3)

4 4 Abstract ad Applied Aalysis It is easily checked that Fially, we will prove I 3 <.Itiseasilyseethat P( P( [X i EX i I( X i a )] >a ε) i= ( X i >a )) I 3 = P( P( i= (EY i EX i I( X i a )) > a ε 2 ) P( X i >a )> ε 2 ) (7) + P( (Y i EX i I( X i a )) >a ε) P( X >a )+ P( (Y i EY i ) > a ε 2 ) + P( (EY i EX i I( X i a )) > a ε 2 ) =I +I 2 +I 3. (4) To prove the desired result (2), it suffices to show I j <for j =, 2, 3. NotethatI <; we oly eed to prove I 2 < ad I 3 <. Note that {Y i EY i,i}are pairwise NQD radom variables by Lemma 4(iii); we have by Markov s iequality, Lemma 4(i), ad the assumptio P( X > a )<that I 2 = C P( C C i= (Y i EY i ) > a ε 2 ) E(Y i EY i ) 2 C EY2 EX2 I( X a )+CP( X >a ) EX2 I( X a )+C. Combiig with (2)ad(5), we have (5) = P(P( X >a )> ε 2 ). I the followig we prove P( X > a ) 0.Notethat P( X > a )<ad 0 P( X > a ) as ;wehave P( X > a ) = o(/), which implies that P( X > a ) 0. Hece, I 3 <. This completes the proof of the theorem. Proof of Theorem 3. We use the same otatios as those i Theorem 2.It is easy to see that P(max S k >a ε) k i= = + P( X i >a ) P(max k P( X >a ) + P( i= P( X >a ) + P( i= k X i I( X i a ) >a ε) X ii( X i a ) >a ε) X ii( X i a ) Y i +Y i EX i I I 2 C =C C C C i= i= i= i=0 i= EX 2 I(a i < X a i )+C EX 2 I(a i < X a i ) =i EX 2 I(a i < X a i )i i ip (a i < X a i )+C P( X >a i )+C<. +C +C (6) C+ P( + P( + P( i= i= i= ( X i a )+EX i I( X i a ) >a ε) a I(X i < a ) a I(X i >a ) > a ε Y i EX i I( X i a ) > a ε EX ii( X i a ) > a ε

5 Abstract ad Applied Aalysis 5 C+ P( + P( + i= i= I( X i >a )> ε Y i EX i I( X i a ) > a ε P(E X I( X a )> a ε =C+J +J 2 +J 3. (8) Similarly, we have J 22 <. Therefore, J 2 <follows by the statemets above. This completes the proof of the theorem. Proof of Theorem 4. Firstly, we will prove that (i) (ii). For fixed, deote Y = a I(X < a )+X I( X a )+a I(X >a ). (22) Similar to the proof of I 2 <i Theorem 2,wehave To prove the desired result (3), it remais to show J i <for i =, 2, 3. By Markov s iequality ad the assumptio P( X > a )<,wehave J C i= P( X i >a )=CP( X >a )<. (9) By the assumptios of Theorem 3 ad Lemma 3, wehave (/a )E X I( X a ) 0, which implies that J 3 <. I the followig, we will prove J 2 <.Itiseasilychecked that J 2 P( i= + P( P( i= + P( Y i EY i > a ε 6 ) i= P(X i < a ) +P (X i > a ) > ε 6 ) (Y i EY i ) + > a ε 2 ) i= (Y i EY i ) > a ε 2 ) + P(P( X >a )> ε 6 ) =J 2 +J 22 +J 23. (20) Similar to the proof of I 3 <i Theorem 2, wecaget that J 23 <. Note that, for fixed, {(Y i EY i ) +,i}ad {(Y i EY i ), i }are both pairwise NQD radom variables. Hece, similar to the proof of I 2 <i Theorem 2,wehave J 2 C C i= EY2 <. E[(Y i EY i ) + ] 2 (2) Var (Y ) EY2 EX2 I( X a )+ P( X >a ) CP( X >a )+C<. =0 It follows by Lemma 2 that sup a i= 2sup 2 i= E Y i EY i a i= E Y i P( X i >a i)+2sup a E X I( X a ) CP( X >a )<. =0 (23) (24) Sice Var(Y )a 2 <foreach,wehaveby(23)ad (24)adLemma 7 that Note that a i= (Y i EY i ) = a i= + a a i= (Y i EY i ) 0 a.s. (25) [X i I( X i a i) EX i I( X i a i)] i= [a i I(X i >a i ) a i I(X i < a i ) a i P(X i >a i ) +a i P(X i < a i )], (26)

6 6 Abstract ad Applied Aalysis ad the assumptio P( X > a ) < implies that I( X >a )<a.s.; we ca get that (a a I(X >a ) a I(X < a ) a P(X >a )+a P(X < a )) I( X >a )+ P( X >a )< a.s., which together with Kroecker s lemma yield that [a i I(X i >a i ) a i I(X i < a i ) a i P(X i >a i ) a i= +a i P(X i < a i )] 0 a.s. (27) (28) By (26)ad(28), we have (X i I( X i a i) EX i I( X i a i)) 0 a.s. (29) a i= It follows by the assumptio P( X > a )<agai that X i I( X i >a i) 0 a.s. (30) a i= Therefore, the desired result (ii) follows by (29) ad(30) immediately. Next, we will prove that (ii) (i). Assume that a i= [X i EX i I( X i a i)] 0 a.s. (3) The, we have X EX I( X a ) a = [X a i EX i I( X i a i)] i= a a a 0 a.s. Note that E X I( X a ) a i= [X i EX i I( X i a i)] = E X (I ( X a N)+I(a N < X a )) a a N a P( X a N )+P(a N < X a ) a N a +P( X >a N ) P( X >a N ) 0, (32) (33) as N, which implies that EX I( X a ) 0 a as. (34) It follows by (32) ad(34) thatx /a 0 a.s., which is equivalet to (i) by Corollary 6.Theproofiscompleted. Proof of Theorem 5. Firstly, we will prove that (ii) (i). It follows by (ii) that X a = a i= X i a a a i= X i 0 a.s., (35) which together with Corollary6 imply that (i) holds. O the other had, assume that P( X > a )<;it follows by Lemma 3 that a EX i I( X i a ) E X I( X a a ) 0 a.s. (36) The desired result (ii) follows by Theorem 4 ad (36) immediately. We have proved that (i) (ii); ext we prove (i) (iii). It follows by Lemma (iii) that, for ay ε>0, P( X >a )< P( X >a ε) < P(X + >a ε) <, P(X >a ε) < P(X + >a )<, P(X >a )<. O the other had, we have proved that (i) (ii); hece, P(X + >a )<, P(X >a )< X + i 0 a.s., i= a X i 0 a.s. i= a X i 0 a.s. i= a (37) (38) Therefore, (i) (iii) follows by the statemets above immediately. This completes the proof of the theorem. Proof of Corollary 6. The techiques used here are the secod Borel-Catelli lemma for pairwise NQD radom variables (seecorollary6)adtheorem 5.Theproofissimilartothat of Corollary 2. of Sug [], so the details of the proof are omitted. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper.

7 Abstract ad Applied Aalysis 7 Ackowledgmets The authors are most grateful to the Editor Agelo Favii ad aoymous referee for careful readig of the paper ad valuable suggestios which helped i improvig a earlier versio of this paper. This work was supported by the Natioal Natural Sciece Foudatio of Chia (2000), the Natural Sciece Foudatio of Ahui Provice (308085QA03), ad the Studets Iovative Traiig Project of Ahui Uiversity ( ). Refereces [] S. H. Sug, O the strog law of large umbers for pairwise i.i.d. radom variables with geeral momet coditios, Statistics & Probability Letters,vol.83,o.9,pp ,203. [2] N. Etemadi, A elemetary proof of the strog law of large umbers, Zeitschrift für Wahrscheilichkeitstheorie ud Verwadte Gebiete,vol.55,o.,pp.9 22,98. [3] X.J. Wag, S.H. Hu, Y. She, ad N.X. Lig, Strog law of largeumbersadgrowthrateforaclassofradomvariable sequeces, Statistics & Probability Letters, vol. 78, o. 8, pp , [4] P.-Y. Che, P. Bai, ad S. H. Sug, O complete covergece ad the strog law of large umbers for pairwise idepedet radom variables, Acta Mathematica Hugarica, vol. 42, o.2, pp ,204. [5] X. Tag, Some strog laws of large umbers for weighted sums of asymptotically almost egatively associated radom variables, Joural of Iequalities ad Applicatios, vol. 203, article 4, 203. [6] K. L. Chug, ACourseiProbabilityTheory, Academic Press, New York, NY, USA, 2d editio, 974. [7] A. Gut, Probability: A Graduate Course, Spriger, New York, NY, USA, [8] E. L. Lehma, Some cocepts of depedece, The Aals of Mathematical Statistics, vol. 37, pp , 966. [9] K. Joag-Dev ad F. Proscha, Negative associatio of radom variables, with applicatios, The Aals of Statistics, vol., o.,pp ,983. [0] P. Matuła, A ote o the almost sure covergece of sums of egatively depedet radom variables, Statistics ad Probability Letters,vol.5,o.3,pp ,992. [] Q.-M. Shao, A compariso theorem o momet iequalities betwee egatively associated ad idepedet radom variables, Joural of Theoretical Probability, vol.3,o.2,pp , [2] Q.Y.WuadY.Y.Jiag, Alawoftheiteratedlogarithmofpartial sums for NA radom variables, Joural of the Korea Statistical Society,vol.39,o.2,pp ,200. [3] X.J. Wag, X.Q. Li, S.H. Hu, ad W.Z. Yag, Strog limit theorems for weighted sums of egatively associated radom variables, Stochastic Aalysis ad Applicatios,vol.29,o.,pp. 4, 20. [4] Q.Y. Wu, A Strog limit theorem for weighted sums of sequeces of egatively depedet radom variables, Joural of Iequalities ad Applicatios, vol.200,articleid383805,8 pages, 200. [5] Q.Y. Wu, A complete covergece theorem for weighted sums of arrays of rowwise egatively depedet radom variables, Joural of Iequalities ad Applicatios, vol.202,article50, 202. [6]X.J.Wag,S.H.Hu,W.Z.Yag,adN.X.Lig, Expoetial iequalities ad iverse momet for NOD sequece, Statistics & Probability Letters,vol.80,o.5-6,pp ,200. [7] X.J. Wag, S.H. Hu, ad W.Z. Yag, Complete covergece for arrays of rowwise egatively orthat depedet radom variables, RevistadelaRealAcademiadeCieciasExactas, Físicas y Naturales A: Matematicas,vol.06,o.2,pp , 202. [8] S. H. Sug, O the expoetial iequalities for egatively depedet radom variables, Joural of Mathematical Aalysis ad Applicatios,vol.38,o.2,pp ,20. [9] C. M. Newma, Asymptotic idepedece ad limit theorems for positively ad egatively depedet radom variables, i Iequalities i Statistics ad Probability, Y.L.Tog,Ed.,vol.5, pp.27 40,IstituteofMathematicalStatistics,Hayward,Calif, USA, 984. [20] X.J. Wag, S.H. Hu, W.Z. Yag, ad X.Q. Li, Expoetial iequalities ad complete covergece for a LNQD sequece, Joural of the Korea Statistical Society, vol.39,o.4,pp , 200. [2] S. H. Sug, Covergece i r-mea of weighted sums of NQD radom variables, Applied Mathematics Letters, vol. 26, o., pp.8 24,203. [22] Q.Y.Wu, CovergecepropertiesofpairwiseNQDradom sequeces, Acta Mathematica Siica,vol.45, o.3,pp , [23] P. Y. Che, Strog law of large umbers for pairwise NQD radom variables, Acta Mathematica Scietia A, vol.25,o.3, pp , [24] C. G. Wa, Law of large umbers ad complete covergece for pairwise NQD radom sequeces, Acta Mathematicae Applicatae Siica,vol.28,o.2,pp ,2005. [25] H. Huag, D. Wag, Q. Wu, ad Q. Zhag, A ote o the complete covergece for sequeces of pairwise NQD radom variables, Joural of Iequalities ad Applicatios, vol. 20, article 92, 20. [26] Y. B. Wag, C. Su, ad X. G. Liu, Some limit properties for pairwise NQD sequeces, Acta Mathematicae Applicatae Siica, vol.2,o.3,pp ,998. [27] R. Li ad W. Yag, Strog covergece of pairwise NQD radom sequeces, Joural of Mathematical Aalysis ad Applicatios,vol.344,o.2,pp ,2008. [28] S.X. Ga ad P.Y. Che, Some limit theorems for sequeces of pairwise NQD radom variables, Acta Mathematica Scietia, vol.28,o.2,pp ,2008. [29] J. H. Shi, O the strog law of large umbers for pairwise NQD radom variables with differet distributios, Acta Mathematicae Applicatae Siica,vol.34,o.,pp.22 30,20. [30] H. Xu ad L. Tag, Some covergece properties for weighted sums of pairwise NQD sequeces, JouralofIequalitiesad Applicatios,vol.202,article255,202. [3] X. Tag, Strog covergece results for arrays of rowwise pairwise NQD radom variables, Joural of Iequalities ad Applicatios,vol.203,article02,203. [32] V. M. Kruglov, A strog law of large umbers for pairwise idepedet idetically distributed radom variables with ifiite meas, Statistics & Probability Letters, vol. 78, o. 7, pp , 2008.

8 Advaces i Operatios Research Volume 204 Advaces i Decisio Scieces Volume 204 Mathematical Problems i Egieerig Volume 204 Joural of Algebra Probability ad Statistics Volume 204 The Scietific World Joural Volume 204 Iteratioal Joural of Differetial Equatios Volume 204 Volume 204 Submit your mauscripts at Iteratioal Joural of Advaces i Combiatorics Mathematical Physics Volume 204 Joural of Complex Aalysis Volume 204 Iteratioal Joural of Mathematics ad Mathematical Scieces Joural of Stochastic Aalysis Abstract ad Applied Aalysis Iteratioal Joural of Mathematics Volume 204 Volume 204 Discrete Dyamics i Nature ad Society Volume 204 Volume 204 Joural of Joural of Discrete Mathematics Joural of Volume Applied Mathematics Joural of Fuctio Spaces Volume Volume Volume 204 Optimizatio Volume Volume 204