The Central Limit Theorems for Sums of Powers of Function of Independent Random Variables

Transcription

1 ScieceAsia 8 () : 55-6 The Ceal Limi Theoems fo Sums of Poes of Fucio of Idepede Radom Vaiables K Laipapo a ad K Neammaee b a Depame of Mahemaics Walailak Uivesiy Nakho Si Thammaa 86 Thailad b Depame of Mahemaics Chulalogko Uivesiy Bagkok 33 Thailad Coespodig auho k_eammaee@homailcom Received Feb Acceped Sep ABSTRACT Le (X ) k k ; be a double sequece of ifiiesimal adom vaiables hich ae oise idepede I his pape e give ecessay ad sufficie codiios fo he ( ( )) ( ( )) ( ) sequece of disibuio fucios of S g X L g X B ( ) o eakly covege o a limiig disibuio fucio F fo each aual umbe ad also fo covegece of (F ) KEYWORDS: ceal limi heoem ifiiely divisible Le vy s fomula Mahemaics Subjec Classificaio: ():6E7 6F5 6G5 INTRODUCTION Le (X ) k k ; be a double sequece of ifiiesimal adom vaiables hich ae oise idepede Le S X L X A hee A ae cosas ad le G be he disibuio fucios of S Necessay ad sufficie codiios fo (G ) o covege o a disibuio fucio G ae ko ad i paicula i is ell ko ha G is ifiiely divisible I 957 Shapio cosideed he limi disibuio () fucios of he sums X L X B hee B () ae suiably chose cosas ad N I Shapio -4 ad Temuipog 5 gave he codiios hich guaaees ha he disibuio fucios of he sums X X L X covege o a limi fo < I 998 Neammaee 6 gave he codiios fo covegece of disibuio fucios of l X l X L l X fo < MAIN OF OBJECTIVE I his ok e coside he disibuio fucios of he sums ( ) S g X g X B ( ) ( ( ) ) L ( ) ( ) hee N ad g: R R saisfies he folloig popeies: (g-) g() (g-) g is coiuous sicly deceasig o ( ] ad sicly iceasig o [ ) (g-3) hee eis posiive cosas ad c such ha g ( ) < cfo all ( ) (g-4) g( ) g( ) Sice g saisfies (g-) ad (g-) e ca ie g( ) if ; g ( ) g ( ) if < hee g : g : R R defied by g () g() ad g : R R defied by g () g() Sice g is coiuous a ad g() e ca assume he i (g-3) has popeies g( ) < ad g( ) < The folloigs ae eamples of g g ( ) c fo c> ad N si if ; g ( ) si if < So Shapio s esuls ae ou special case ( ) ( ) Fom o o fo N e le F F F be ( ) ad ( he disibuio fucios of S ) g( X ) X especively ad fo ifiiely divisible disibuio fucio F e le M N γ σ be M N γ σ i

2 56 ScieceAsia 8 () Le vy s fomula of F (Peov 7 chape II) The ecessay ad sufficie codiios fo covegece ( of he sequece of disibuio fucios of S ) ad he sequece of disibuio fucios F ae give i Theoem A ad Theoem B hich saed belo Theoem A Assume ha G G as The fo each N ad fo suiably chose ( cosas B( ) F ) F as if ad oly if k g lim limsup { ( g ( )) df ( ) ( g ( )) df ( ) ad k g g ( ( g ()) df ( ) ( g ()) df ( )) } σ < g k g lim lim if { ( g ( )) df ( ) ( g ( )) df ( ) k g g ( ( g ()) df ( ) ( g ()) df ( )) } < g σ ( ) Theoem B Le G G ad F F as fo all N The F H ad if ad oly if M( ) < g fo all ( ) N( ) > g fo all ( ) 3 lim σ ( σ ) hee he fucios M N ae fucios i Le vy s fomula of F ad σ is he cosa i Le vy s fomula of H Moeove e ko ha 4 if σ M is coiuous a g () ad N is coiuous a g () he H is degeeae 5 if σ M is coiuous a g () ad N is coiuous a g () he H is omal 6 if σ M is discoiuous a g () o N is discoiuous a g () he H ( - m) is Poisso fo some cosa m 7 if σ M is discoiuous a g () o N is discoiuous a g () he H is he disibuio fucio of he sum of o idepede adom vaiables oe of hich is omal ad he ohe is Poisso PROOFS OF MAIN RESULTS Befoe e pove he mai esuls e eed he folloig lemmas Lemma Le X ~ N(a σ ) ad Y ~ Poi(λ) If X ad Y ae idepede he Le vy s fomula of he chaaceisic fucio of X Y is λ i i log ϕ X Y( ) i( a ) σ ( e ) dk( ) hee K : R R is defied by K λ if < ; ( ) if > Poof Le ϕ X ad ϕ Y be he chaaceisic fucios of X ad Y especively Fom Lukacs 8 p93 e have log ϕx ( ) ia λ i i σ ad log ϕy ( ) i ( e ) dk( ) Sice X ad Y ae idepede log ϕ ( ) log ϕ ( ) ϕ ( ) X Y X Y log ϕ ( ) log ϕ ( ) X ia λ i i e i σ ( ) dk( ) Y λ ia i e i ( ) σ ( ) dk( ) # Lemma If G G as he fo evey N k ( ) lim F ( ) fo all < ad k k k ( ) lim ( F ( ) ) N( g ( )) M( g ( )) ae o ( ) ( Fuhemoe if F ) F fo evey N he fo each N e have 3 M o ( ) ad 4 N ( ) N( g ( )) M( g ( )) ae o ( ) hee M ad N ae fucios i Le vy s fomula of F

3 ScieceAsia 8 () 57 Poof Noe ha F ( ) if < ; ( ) PX ( ) if ; ad F F ( g ( )) F ( g ( ) ) if > ( ) if < ; ( ) ( ) F ( ) if () () So follos fom () To pove le N Sice G G by Theoem 8 of Peov 7 p8-8 e ko ha k k lim F ( ) M( ) ad lim ( F ( ) ) N( ) (3) k k fo all coiuiy pois of M ad N Fom () ad (3) k ( ) k lim ( F ( ) ) k lim { F ( g ( )) F ( g ( ) )} k k lim { F ( g ( )) } lim { F ( g ( ) ) } k k k N( g ( )) M( g ( ) ) ae o ( ) N( g ( )) M( g ( )) ae o ( ) ( No e suppose ha F ) F fo evey N By () () ad Theoem 8 of Peov 7 p8-8 e have (3) ad (4) # ( Lemma 3 Assume ha F ) F fo evey N The fo evey N M( ) o ( ) ad N( ) N( ) ae o ( ) Poof We use he same agume i povig of Lemma by usig () isead of () # Lemma 4 Assume ha ( ) fo evey N F F as ad F H as The H is oe of he folloig a degeeae disibuio fucio a Poisso disibuio fucio 3 a omal disibuio fucio 4 he disibuio fucio of he sum of o idepede adom vaiables oe of hich is omal ad he ohe is Poisso Poof Le be ay aual umbe The by Lemma 3 e have M o (- ) ad N( ) N ( ) ae o ( ) Sice F H as by Theoem 3 of Peov 7 p75 e have lim M ( ) M ( ) fo all coiuiy pois of M lim N ( ) N ( ) fo all coiuiy pois of N limγ γ ad lim limsup udm( u) σ udn( u) lim limif ( ) ( ) udm σ ( σ ) u udn u hee M N γ ad σ ae associaed ih H i Le vy s fomula This shos ha M N N N ( ) if ; ( ) lim ( ) > ad N( ) if < < Bu if > ; N ( ) so N( ) Thus N ( ) N( ) if < < Case σ ad N The H is degeeae Case σ ad N The H is omal Case 3 σ ad N akes oe jump If γ ( ) N he H is Poisso

4 58 ScieceAsia 8 () N ( ) γ N ( ) If γ le m e oe ha he chaaceisic fucio ϕ () m of H ( m ) im is e ϕ () hee ϕ is he chaaceisic fucio of H Hece log im ϕ ( ) log e ϕ ( ) m im log ϕ ( ) i i im iγ ( e ) dn ( ) N( ) i i i( ) ( e ) dn ( ) So H( - m) is Poisso Case 4 σ ad N akes oe jump By Lemma H is he disibuio fucio of he sum of o idepede adom vaiables oe of hich is a Poisso ad he ohe is a omal # ( ) Lemma 5 Assume ha F F as fo evey N ad G G as If F H as he M o ( ) if > ; N ( ) N( g ( )) M( g ( )) if < < o ( ) ad 3 M( g ( )) N( g ( )) hee M ad N ae fucios i Le vy s fomula of F ad Mad N ae fucios i Le vy s fomula of H Poof Use he same echique i fidig N ad M i Lemma 4 by usig Lemma isead of Lemma 3 # Poof of Theoem A Noe ha fo > e have ( ) ( ) df ( ) ( df ( )) < < d[ F ( g ( )) F ( g ( ) ) ] d[ F g F g ] ( ( ( )) ( ( ) ) ) g ( g ( )) df ( ) ( g ( )) df ( ) g g g ( ( g ( )) df ( ) ( g ( )) df ( ) ) g [ g ( ) ad g ( )] ( g ( )) df ( ) ( g ( )) df ( ) g g ( ( g ()) df () ( g ()) df ( )) g To pove ecessiy e suppose ha F (4) ( ) as The ad follo fom Theoem 8 of Peov 7 p8-8 ad (4) Fo sufficiecy e defie M :( ) R ad d N :( ) R by M ( ) ad N ( ) N( g ( )) M( g ( )) Clealy M ad N ae odeceasig ad M (- ) N ( ) By () ad () of Lemma e k k ( ) have lim F ( ) M ( ) ad (5) k ( ) lim ( F ( ) ) N( ) (6) k fo all coiuiy pois of M ad N By assumpios ad (4) e have k ( ) ( ) lim limsup df ( ) ( df ( )) k < < { } k ( ) ( ) lim limif { df ( ) ( df ( )) } σ < k < < F (7) By (5)-(7) ad Theoem 8 of Peov 7 p8-8 F F as # ( ) Poof of Theoem B Fo ad < < mi {(g ()) (g (-)) } e have ma{ g g } ad udm( u) udn( u)

5 ScieceAsia 8 () 59 ud[ N( g ( u)) M( g ( u)) ] (by Lemma (3) ad(4)) g ( ) g ( ) ( g ( )) dn( ) ( g ( )) dm( ) g ( ) [ g ( u ) ad g ( u )] ( g ( )) dn( ) ( g ( )) dm( ) g ( ) g ( ) ( ( g ()) dn()) ( ( g ()) dm()) g ( ) { ( g ()) dn() ( g ()) dm()} { ( g ()) dn() ( g ()) dm()} { g () () g () dn dm()} c { dn( ) dm( )} The Hece (by popey (g-3))(8) lim limsup { udm( u) udn( u) } lim limsup c { dn( ) dm( )} { } lim limsup udm( u) udn( u ) (9) Similaly e have lim limif udm( u) udn( u ) () { } To pove ecessiy e suppose ha F H as Sice G G by Theoem 8 of Peov 7 p8-8 k k e have lim F ( ) M( ) ad lim [ F ( ) ] N( ) k k fo all coiuiy pois of M ad N The () ad () follo fom Lemma 5(3) ad he fac ha M ad N ae odeceasig ad M(- ) N( ) No e ill sho (3) Sice F H by Theoem 3 of Peov 7 p75 e have lim limsup { udm( u) σ udn( u) } lim limif udm( u) σ udn( u) ( σ ) { } By (9) - () e see ha limsup σ ( σ ) ad limif σ ( σ ) So lim σ ( σ ) () To pove sufficiecy e assume ha () () ad (3) hold ( ) Sice G Gad F F as by Lemma M ad N( ) N( g ( )) M( g ( )) ae o ( ) Le N : R R be defied by N ( ) lim N ( ) ) M : ad R R be defied by M ( ) lim M( ) The M o (- ) ad by assumpios () ad if > ; () N ( ) N( g ( )) M( g ( )) if < < o ( ) Tha is M (- ) N( ) Fom assumpio (3) ad (9) e have lim limsup udm( u) σ udn( u ) lim σ ( σ ) { } Similaly e ca sho ha lim limif udm( u) σ udn( u ) ( σ ) { } By Theoem 3 of Peov 7 p75 e have lim F ( ) H( ) hee H is he ifiiely divisible disibuio deemied by M N γ ad (σ) By he same agume of Lemma 4 e have (4)-(7) # REFERENCES Shapio JM (957) Sums of Poes of Idepede Radom Vaiables The Ameica Mahemaical Sociey Shapio JM (975) Domai of aacio ecipocals of poe of adom vaiables Siam Joual Appl Mah

6 6 ScieceAsia 8 () 3 Shapio JM (977) O domias of omal aacio o sable disibuios Houso J Mah Shapio JM (988) Limi disibuios fo sums of ecipocals of idepede adom vaiables Houso J Mah Temuipog I (986) Limi Disibuios fo Sums of he Recipocal of a Posiive Poe of Idepede Radom Vaiables Ph D hesis Chulalogko Uiv 6 Neammaee K (998) Limi Disibuios fo Radom Sums of he Recipocals of Logaihms of Idepede Coiuous Radom Vaiables J Sci Res Chula Uiv 3 No 7 Peov VV (975) Sum of Idepede Radom Vaiables Spige-Valag NeYok 8 Lukacs E (97) Chaaceisic Fucios Hafe Publishig Compay NeYok