ON THE LAWS OF LARGE NUMBERS FOR DEPENDENT RANDOM VARIABLES

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1 Joural of Sees Islam Republ of Ira 4(3): 7-75 (003) Uversty of Tehra ISSN ON THE LAWS OF LARGE NUMBERS FOR DEPENDENT RANDOM VARIABLES HR Nl Sa * ad A Bozorga Departmet of Mathemats Brjad Uversty Brjad Islam Republ of Ira Departmet of Statsts Faulty of Sees Ferdows Uversty Mashhad Islam Republ of Ira Abstrat I ths paper we eted ad geeralze some reet results o the strog laws of large umbers (SLLN) for parwse depedet radom varables [3] No assumpto s made oerg the estee of depedee amog the radom varables (heeforth rv s) Also Chadra s result o Cesàro uformly tegrable rv s s eteded Keywords: Complete overgee; Strog law of large umbers; Parwse egatvely depedet rv s; Negatvely assoated rv s; Cesàro uformly tegrable rv s Let Itroduto ad Prelmares { X } be a sequee of tegrable rv s defed o the same probablty spae ad put S( ) X X S( )/ Laders ad Rogge [8] proved a strog law of large umbers (SLLN) for parwse depedet ad strogly uformly tegrable rv s Chadra ad Goswam [3] proved a more geeral SLLN for parwse depedet ad Cesàro uformly tegrable rv s Laders ad Rogge [9] showed that Chadra s results hold for o-egatve ad uorrelated stead of parwse depedet rv s but ot wthout the assumpto of o-egatvty Matula [0] has proved the SLLN for parwse egatvely depedet rv s wth the same dstrbuto Bozorga et al [] obtaed the SLLN for weghted sums of a array of rowwse egatvely depedet rv s uder erta momet odtos Am [] has proved the SLLN for speal egatvely depedet rv s ad for weghted sums of uformly bouded egatvely depedet rv s I ths paper we modfy ad geeralze some theorems of SLLN of Chadra ad Goswam [3] for parwse egatvely depedet rv s whh are ot eessarly detally dstrbuted Defto The radom varables X X ( ) are sad to be parwse egatvely depedet (heeforth parwse ND) f () P ( X > X > ) P ( X > ) P ( X > j j j j ) for all j R j It a be show that () s equvalet to () P ( X X ) P ( X ) P ( X j j j j ) for all R j j Defto ([7]) The radom varables X X ( ) are sad to be egatvely assoated (NA for short) f for every par of dsjot oempty subsets A A of { } (3) Cov f X A f X A ( ( ) ( )) 0 * E-mal: l@mathumar 7

2 Vol 4 No 3 Summer 003 Nl Sa ad Bozorga J S I R Ira wheever f ad f are oordatewse reasg suh that ths ovarae ests Clearly (3) holds f both f ad f are dereasg A fte olleto of { X } s sad to be parwse ND (egatvely assoated) f every fte subolleto s parwse ND (egatvely assoated) It a be show that NA mples parwse ND ad for ND s equvalet to NA Ma Results I ths paper C stads for a geer ostat ot eessarly the same at eah appearae Also { f ( )} wll stad for a reasg sequee suh that f ( ) > 0 for eah f ( ) ad for > m( ) log f ( ) the teger part of log f ( ) s a reasg sequee I the followg Theorem we preset aother poof for the theorem of Csörgo et al [4] (see Chadra ad Goswam [3]) Theorem Let { X } be a sequee of rv s wth fte Var( X ) Assume that ) there s a double sequee { ρ } of o-egatve reals suh that Var( S( )) ρ for eah ; j ) ρ /( ( )) f j < j ma( j) The j [ S( ) E( S( ))]/ f ( ) 0 ompletely the sese of Hsu ad Robbs [6] (see also page 5 of Stout []) Proof Put Z ( ) ( S( ) ES( )) It s suffet to f ( ) show that P( Z( ) > ε) < P( Z ( ) > ε ) CE ( Z ( )) C E( S( ) ES( )) f ( ) ρ C C ρ j f ( ) j j f ( ) The relato m ( ) [log f ( )] ow mples m ( ) m ( ) + f ( ) < ad Thus the last sum s C ρ j f ( ) f ( j) m( ) j m( ) + f ( j) m( ) f ( ) m( ) C ρ m ( ) + Let P f{ f ( j )} The the RHS above s m( ) C ρ j m C P ρ j m m( P) m( P) C ρ C j j ρ < f ( j) The [ S( ) E( S( ))]/ f ( ) 0 ompletely ad the Borel-Catell lemma mples that [ S( ) E( S( ))]/ f ( ) 0 as Proposto ([]) Let{ X } be a sequee of parwse ND rv s If { f } s a sequee of mootoe reasg (or mootoe dereasg) futos the { f( X ) } s a sequee of parwse ND rv s Corollary Let { X } be a sequee of parwse ND rv s The { + X } ad { X } are two sequees of parwse ND rv s where X + ad X are postve ad egatve parts of a radom varable X respetvely Now we are able to prove the followg theorems for parwse ND radom varables wth fte varaes Theorem Let { X } be a sequee of parwse ND rv s wth fte Var( X ) Assume that ( f ( )) Var( X ) < 7

3 J S I R Ira Nl Sa ad Bozorga Vol 4 No 3 Summer 003 The S( ) E( S( )) / f ( ) 0 ompletely Proof Uder parwse ND odto we have Var( X ) Var( X ) ρ j j where ρ Var( X ) for j ad ρ 0 for j It follows from Theorem that ompletely Eample Let { X } S( ) E( S( )) 0 f ( ) be a sequee of d radom varables wth fte Var( X ) ad f ( ) > It s obvous that odtos of Theorem hold S( ) E( S( )) ad we have 0 ompletely f ( ) Eample Let { X } ad f ( ) be as above Y a X a >0 ad a O( β ) β > 0 Put Z X Z Y ad S( ) Z It s obvous that { Z s a sequee of parwse ND rv s } wth fte Varaes Also ( f ( )) Var( Z ) ( f ( )) Var( Z ) + ( f ( )) Var( Z ) ( f ( )) Var( X ) ( f ( )) avar( X ) < + The by Theorem S( ) E( S( )) 0 f ( ) ompletely The et theorem s a aalogue of the three-seres theorem of Kolmogorov (99) for depedee rv s Our teto s to replae the odtos of Chadra ad Goswam [3] by sutable weaker odtos of smple ature (see ths oeto page 8 of Chug [5]) Theorem 3 Let { X } be a sequee of parwse ND tegrable rv s suh that there s a sequee { B } of Borel subsets of R that are sem tervals ( ] (( ) [ ) or ( )) satsfyg the followg odtos: (a) CP( X B) < where C ( / f ( )) ; (b) E( X I( X B )) o( f ( )) ; () ( f ( )) E( X I( X B )) < ; here B s the omplemet of B The S () ES (())/ f() 0 almost surely as Proof Let Y X I( X B) + I( X B) By Proposto { Y } s a sequee of parwse ND rv s We use Theorem for { Y } ( f ( )) Var( Y ) f ( ) E( Y ) f ( ){ X dp( w ) + P( X B )} X B f ( ) E( X I( X B )) + P( X B) f ( ) f ( ) E( X I( X B )) + C P( X B ) < the Theorem appled to { Y } yelds ( Y E( Y )) 0 f ( ) as It s easy to show that ( Y E( X )) ( Y E( Y f ( ) f ( ) )) + PX ( B ) f ( ) 73

4 Vol 4 No 3 Summer 003 Nl Sa ad Bozorga J S I R Ira Se C : ( E( X I( X B )) f ( ) PX ( B) f ( ) P( X B) + P( X B) f ( ) f ( ) C : C P( X B ) < the by Kroeker s lemma we have P( X B ) f ( ) 0 By (b) we get ( Y E( X)) 0 f ( ) Se by (a) rv s { X } ad { Y } are equvalet the by the frst Borel-Catell lemma the desred result follows I the et theorem we use the followg lemmas Lemma a be proved usg the summato by parts formula ad Lemma s Lemma 5 of Petrov ([] 77-78) Lemma If b < ad b s dereasg the for ay bouded { } suh that { } s reasg ( ) b < We deote by ψ the set of futos ψ ( ) suh that (a) ψ ( ) s postve ad o-dereasg the terval > 0 for some 0 ad (b) the seres / ψ ( ) overges Lemma Let { a } be a sequee o o-egatve umbers A a A The the seres a / Aψ ( A) overges for ay ψ ψ We et geeralze the SLLN of Chadra ad Goswam [3] The reader should ote the aturalty of Cesàro uform tegrablty the otet of laws of large umbers Theorem 4 Let { X } be a sequee of parwse ND rv s Assume that there s a futo Φ : (0 ) (0 ) suh that ) f Φ ( ) / > 0 ; ) t Φ() t s reasg to as t ; ) ( Φ ( )) <; v) sup[ E( Φ ( X ))] ( say) < The ( X E( X)) f ( ) 0 almost surely as Proof We use Theorem 3 wth B ( ] for It s easy to hek that C M < for eah Put [ E( Φ( X ))] for We frst verfy Codto (a); C P( X B ) M P( Φ( X Φ( )) M E( Φ( X )/ Φ ( )) < by Lemma ad (v) To prove Codto (b) let ε > 0 There s a teger N > suh that for eah ( t+ ) Φ() t for t > N ad so for eah ε ( ( > )) E X I X N ( ( ))/( ) ε ε E Φ X I X > N + < / Net there s a teger N N N > N suh that for eah E( X ) < ε / The for N E( X I( X > )) E( X I( X > )) N E( X ) + E( X I( X > N)) < ε It s lear that B C D E where 74

5 J S I R Ira Nl Sa ad Bozorga Vol 4 No 3 Summer 003 C ( ) /4 /4 D [ ] [ ] ad E ( ) To prove Codto () t suffes to show that /4 /4 (5) E( X I( X C )) < ad E XI X E ( ( )) < E XI X D ( ( )) < We frst show that E( X I( X C )) < Se f{ y : y Φ ( ) / < } s postve the there s a z the terval ( ) suh that / z f{ y : y Φ ( )/ < } Hee we have / z z Φ( ) Hee E( X I( X < ) ) Φ ( )/ for eah < ad z E( Φ( X ))< To omplete the proof that Codto () holds t suffes to show that (6) E( X I( X D )) < For eah there s a z the terval [ ] suh that / z /4 / 4 f{ y : y Φ( )/ : } ote that the rght sde of the above equalty s postve The for [ / 4 ] we have Φ( ) z (as z ) Φ ( )/ t (by z ad( )) 3/4 /4 /4 where t Φ ( ) for Observe that Φ E( X I( X D )) E( ( X ))/ t t [ ( ) ]/ So (5) wll follow f we show that / t < (usg Lemma ) For ths purpose we use Lemma /4 /4 wth a ( ) for ψ ( ) Φ ( )/ ; here we are followg the otato of Petrov [] ad usg Assumptos () ad () As 3/4 /4 /(4 ) for eah ad t ψ ( ) we get a / t < Proposto If { X } s a sequee of NA rv's the Theorems 3 4 are vald Akowledgemets The authors wsh to thak the referees ad the edtor for ther useful ommets ad suggestos Referees Am M PhD Thess Mashhad Uversty (000) Bozorga A Patterso RF ad Taylor RL Lmt Theorems for depedet radom varables World Cogress Nolear Aalyss 9: (996) 3 Chadra TK ad Goswam A Cesàro uform tegrablty ad the strog law of large umbers Sakhya: The Ida Joural of Statsts 54A(): 5-3 (99) 4 Csörgo S Tador K ad Totk V O the strog law of large umbers for parwse depedet radom varables Ata Math Hugara 4: (98) 5 Chug KL A Course Probablty Theory Seod edto Aadem Press New York (974) 6 Hsu PL ad Robbs H Complete overgee ad the law of large umbers Pro Nat Aad S USA 33: 5-3 (974) 7 Joag-Dev K ad Prosha F Negatve assoato of radom vaables wth applatos A of Stat : (983) 8 Laders D ad Rogge L Laws of large umbers for parwse depedet uformly tegrable radom varables Math Nahr 30: 89-9 (986) 9 Laders D ad Rogge L Laws of large umbers for uorrelated Cesàro uformly tegrable radom varables Sakhya: The Ida Joural of Statsts 59A(3): (997) 0 Matula P A ote o the almost sure overgee of sums of egatvely depedet radom varables Stat Probab Letters5: 09-3 (99) Petrov VV Sums of Idepedet Radom Varables Sprger-Verlag New York-Hedelberg-Berl (975) Stout WF Almost Sure Covergee Aadem Press New York (974) 75