Complete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables

Transcription

1 Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl Sa, ad A. Bozorga Departmet of Statstcsaculty of Mathematcal Sceceserdows Uversty of Mashhad, Mashhad, Islamc Republc of Ira Departmet of Statstcs, Uversty of Brad, Brad, Islamc Republc of Ira Abstract Let } be a sequece of arbtrary radom varables wth E 0 ad E <, for every ad {a } be a array of real umbers. We wll obta two maxmal equaltes for partal sums ad weghted sums of radom varables ad also, we wll prove complete covergece for weghted sums a, uder some codtos o a ad sequece, }. Keywords: Complete covergece; Weghted sums; Maxmal equaltes; Par-wse egatve depedece. Itroducto The cocept of complete covergece of a sequece of radom varables was troduced by Hsu ad Robs [4], as follows. A sequece, } of radom varables coverges completely to a costat a (deoted lm a, completely), f P[ - a > ] < for all > 0. From the o, there are may authors who devote the study to the complete covergece for partal sums ad weghted sums of depedet radom varables such as Taylor [], Hu et al. [6], Sug et al. [], Wedeg ad Zhegra [5] ad Sug []. Several authors exteded ths covergece to partal sums ad weghted sums of egatvely depedet ad egatvely assocated radom varables amely Lag ad Su [0], Lag [9], Huag ad u [5] ad Am, ad Bozorga []. I ths paper frst, we prove two maxmal equaltes for partal sums ad weghted sums of arbtrary radom varables ad the preset varous codtos o { a } ad } for whch I addto we cosder a coverges completely., } as a sequece of radom varables wth zero meas such that + x -γ t P[ x] M e dt, () for all ad all x 0, where M ad γ are postve costats. Haso ad Wrght [], obtaed a boud o tal probabltes for quadratc forms depedet radom varables usg codto (). Wrght [6] proved that the boud establshed by Haso ad Wrght [] for depedet symmetrc radom varables also holds whe the radom varables are ot symmetrc but codto () s vald. * Correspodg author, Tel.: +98(5)888605ax: +98(5)888605, E-mal: m-am@ferdows.um.ac.r

2 Vol. 8 No. 4 Autum 007 Am et al. J. Sc. I. R. Ira Defto. ([]) The sequece, } of radom varables s sad to be par-wse egatve depedet (PND) f for every x, x R P[ x, x ] P[ x ]P[ x ]. Lemma. ([]) If the sequece the, } s PND, E(. ) E( ).E( ) for all. Lemma. If Z~N(0, ), ad satsfes (), the there exsts λ such that E λ EZ. Proof. By codto (), we get + E xp[ > x]dx 0 Where M x 0 x γt ( e dt) γt 0 M t e dt λ EZ, 4 M π λ <. 4 γ Lemma. ([8]) The sequece almost surely f ad oly f lm P[sup > ] 0, for every > 0. +, } coverges Theorem. ([7]) If {( ), } s a oegatve sub-martgale, the p p p p E (max ) ( ) E, whe p >. 0 p.the Maxmal Iequaltes I ths secto, we prove two maxmal equaltes ad exted Kolomogorov s covergece crtero of strog law of large umbers ad we obta some other useful results. Theorem. Let radom varables wth, } be a sequece of E( ) 0, E <,, the for the gve > 0, P[ max S ] ( ) Where S () ad var( ). Proof. Set S + ad S, + where max{0, } ad max{0, -}. Sce E[S F - ] S (-), a.e. ad E[S F - ] S (-), a.e. hece the sequeces {S, } ad {S, } are oegatve sub-martgales where F (,, ) for all, where (,,..., ) s the smallest sgma fled produced by,,...,. The, we get by Marov s equalty ad Theorem for p, that P[ max S > ] E[max S ] 4ES 4 ( ) for all > 0, the last equalty s true by the followg statemet + ES ( ). Smlarly, oe ca show that 4 P[ max S > ] ( ). Combg these two equaltes ad S S + S, we obta P[ max S ] P[max S + P[max S ( ) for all 0, > The followg corollary s a exteso of Kolomogorov s covergece crtero of strog law of large umbers for arbtrary radom varables. Corollary. Let, } be as Theorem. ) If <, the the seres

3 Complete Covergece ad Some Maxmal Iequaltes for coverges a.e. ) If hold. ad b b <, the the followg statemets 0 a. e. as, () S E sup < forall 0 < b where {b } s a sequece of postve creasg real umbers such that b as. (4) Proof. ) By applyg Lemma, Theorem ad Lemma, we have P[sup S - S > ] Sce + lm P[ sup S - S > ] m m + ( ), for all 0, + > m <, t follows that lm P[ sup S - S > ] 0, + ths completes the proof. ) Tag Y, we get () ad (4) by b Keroecer s Lemma, Lemma ad Theorem. Corollary. Let, } be as Theorem. ) If <, the for the gve > 0 ad for > 0, the followg statemets hold ad P S [max > ] <, (5) S P[sup > ] <, (6) ) If ( ) <, for some 0 the for every > 0, >, P[ max S > ] < for all > 0. (7) + I partcular, f ( ) O, for some > 0 ad >, the we ca obta (5). ) If } satsfes codto (), the O(), ad for 0 < <, we have ( ) < Theorem. Let, } be a arbtrary sequece of radom varables wth E( ) 0, E <,. Suppose that {a,, } be a array of real umbers, the P[ max T > ] where T a. ( a ) for all > 0, (8) + Proof. Set T a ad T a. Sce E[T F - ] T(-), a.e. ad E[T F ] T, a.e. t follows that the sequeces - (-) {T, } ad {T, } are oegatve sub-martgales, where F (,, ) for all. Sce T T + T for all, the proof of (8) follows from the same argumet as that the proof of Theorem. Hece P[ max T ] P[ max T + P[ max T ( a ) for all 0. > Corollary. Let, } ad {a } be as Theorem, ) If ( a ) <, for some > 0, the we have P T forall [max > ] < > 0. (9)

4 Vol. 8 No. 4 Autum 007 Am et al. J. Sc. I. R. Ira ) If } satsfes codto () ad O( ) for all > 0, the (9) holds. a.complete Covergece for Weghted Sums Usg results of Secto, we obta complete covergece for weghted sums a of radom varables that satsfy () uder some codtos o a. Theorem 4. Let of radom varables ad array of real umbers such that (+ ), } be a arbtrary sequece {a,, } be a a - a O( ) for some > 0, where a( + ) 0. ) If, } satsfes codto (), the a 0, completely as. (0) ) If O( ) for some > 0, the (0) holds. Proof. Usg Abel s partal summato rule we get S (+ ) a max ( a a ) C max S for some > 0. From Theorem, we coclude that P( max S > ) - for all 0. > Hece () ad () yeld - P[ a > ] P( max S > ) ) Codto () mples that -. () () λ < > for all /. ) If O( ) for some > 0, the < M < for all > +. Where 0 < M <, ths completes the proof. Theorem 5. Let, } be a arbtrary sequece of radom varables wth zero meas that satsfes () ad { a } be as Theorem 4. The, there exsts λ > 0 such that λ P[ a > ] a for all > 0. Proof. Applyg Marov s equalty, Cauchy- Shwarz s equalty ad Lemma, we have P[ a > ] E a ( a ) E + a a E ( a ) E + a a E E a λ a for all > 0. Corollary 4. Let, } ad { a } be as Theorem 5. ) If a O(), the for all >, a 0, completely as. 4

5 Complete Covergece ad Some Maxmal Iequaltes for ) If max a O( ), 0 < δ <, the δ for all > -, we have - δ - a 0, completely as. Theorem 6. Let, } be a sequece of PND radom varables wth zero meas that satsfes (). Let { a } be a array of postve real umbers wth a ( ), O δ for all δ 0 + δ >, we have - a 0, completely as. The for all. Proof. By Marov s equalty, Lemmas ad, we get P[ a > ] E a + a E a a E E λ a λ < for all > 0. δ Ths completes the proof. 4. Examples I the followg we have several examples that satsfy the codtos of Theorem, ad Corollares ad.. Let, } be a sequece of arbtrary radom varables. ) If ~exp( λ ) for all ad <, λ the coverges a.e. I partcular f λ for all >, the coverges a.e. ) Let P[ 0] - for all > 4. Sce ad P[ ] <, t follows that coverges a.e. ) If ~U(-a, a ), 0 < a < for all, ad a <, the coverges a.e. v) Let ~ Γ (m 0, ), for all > 0. Sce <, t follows that ( - E ) 0 a.e.. Let } be a sequece of..d. radom varables wth dstrbuto U[0, ]. Set Y ad Var(Y ). It s obvous that EY ad V ar(y ) <, for all. Sce <, from Corollary., we coclude that ( Y ) EY coverges a.e. Next, ote that EY < a.e. hece, coverges a.e. Also, the codtos of Corollary. are vald for the sequece {Y }. Thus (5) ad (6) hold.. Let } be a sequece of radom varables wth the probablty fucto, P[ 0] P[ ] for all. It s obvous that codtos of Corollares ad are vald for the sequece { }, hece coverges a.e. ad the statemets (5) ad (6) are true. Acowledgmets The authors are extremely grateful to the referees for ther valuable commets ad suggestos that led to a remarable mprovemet of the paper. Supported from 5

6 Vol. 8 No. 4 Autum 007 Am et al. J. Sc. I. R. Ira "Ordered ad Spatal Data Ceter of Excellece of Ferdows Uversty of Mashhad" s acowledged. Refereces. Am M. ad Bozorga A. Complete covergece for ND radom varables. Joural of App. Math. ad Stoch. Aalyss., 6(): -6 (00).. Bozorga A., Patterso R.F., ad Taylor R.L. Lmt theorems for depedet radom varables: World Cogress Nolear Aalysts 9. p (996).. Haso D.L. ad Wrght F.T. A boud for tal probabltes for quadratc forms depedet radom varables. A. Math. Statst., 4: (97). 4. Hsu P.L. ad Robbs H. Complete covergece ad the law of large umbers. Proc. Nat. Acad. Sc., : 5- (947). 5. Huag W.T. ad u B. Some maxmal equaltes ad complete covergece of NA radom sequeces. Statstcs & Probablty Letters, 57: 8-9 (00). 6. Hu T.C., Szyal D., ad Volod A.I. A ote o complete covergece for arrays. Ibd., 8: 7- (998). 7. Gut A. Probablty: A Graduate Course. Sprger, New Yor (005). 8. Karr F.A. Probablty. Sprger Verlage, New Yor (99). 9. Lag H.Y. Complete covergece for weghted sums of egatvely depedet radom varables. Statstcs & Probablty Letters, 48: 7-5 (000). 0. Lag H.Y. ad Su C. Complete covergece for weghted sums of NA sequeces. Ibd., 45: (999).. Sug S.H., Volod A.I., ad Hu T.C. More o complete covergece for arrays. Ibd., 7: 0- (005).. Sug S.H. Complete covergece for weghted sums of r.v. s. Ibd., 77: 0- (007).. Taylor R.L. Complete covergece for weghted sums of array of radom elemets. J. Math. I & Math. Scece., 6(): (98). 4. Techer H. Almost certa covergece double array. Z.Wahvch. Ver.Gebete., 69: -45 (985). 5. Wedeg L. ad Zhegra L. A supplemet to the complete covergece. Statstcs & Probablty Letters., 76: (006). 6. Wrght F.T. A boud o tal probabltes for quadratc forms depedet radom varables whose dstrbutos are ot ecessarly symmetrc. The Aals of Probablty, (6): (97). 6