Sherzod M. Mirakhmedov,

Transcription

1 Approxao by ora dsrbuo for a sape su sapg whou repacee fro a fe popuao Ibrah B Mohaed Uversy of Maaya Maaysa ohaed@ueduy Sherzod M Mrahedov Isue of Maheacs Tashe shrahedov@yahooco Absrac A su of observaos derved by a spe rado sapg desg fro a popuao of depede rado varabes s suded A procedure fdg a geera er of Edgeworh asypoc expaso s preseed The Ldeberg codo of asypoc oray Berry-Essee boud Edgeworh asypoc expasos uder weaeed codos ad Craer ype arge devao resus are derved Key words ad phrases: Berry-Essee boud Edgeworh expaso Ldeberg codo arge devao fe popuao sape su sapg whou repacee MSC (000): 6G0 60F05 Iroduco ( Y Y ) be a popuao of depede rado varabes (rv) ad r ( r r ) Le vecor (rvec) depede of Y Y ad such ha Pr r of he ubers Le r r be a rado! for ay peruao S Y Y ha s a su of rvs chose a rado whou repacee fro he popuao We are eresed he approxao of he dsrbuo of S by a ora dsrbuo The sape su S s of eres he coex of s sasca appcaos Uder he hypohess of hoogeey of wo sapes he wo-sape ear ra sascs ca be reduced o ha of sape su S whe he eees of he popuao are o-rado rea ubers; hs case he eees of he popuao becoe he scores of he ra sasc see Bce ad vo Zwe (978) Suao where Y are rado varabes arses for sace he probes assocaed wh wo-sage sapes parcuar srafed rado sapg a ear esao for he ea oa of a srafed popuao see Cohra (96) oe aso f he S spy s a su of depede rvs May auhors have suded he sape su I he suao whe Y are o-rado a suffce ad ecessary codo sar o he we-ow Ldeberg codo for asypoc oray of S have bee preseed by Erdös ad Réy (959) ad Hae (960); we shoud refer aso o Wad ad Wofowz (9) who for he frs e have proved asypoc oray of S uder cera codos The rae he cera heore (CLT) was obaed by Bes (969) ad Högud (978) The secod order approxao resus was derved by Robso (978) Bce ad va Zwe (978) Babu ad Sgh (985) Babu ad Ba (996) ad Bozes (000) Aso Craer s arge devao resu

2 has bee proved by Robso (977) ad Hu e a (007b) The case whe he eees of he popuao are rvs has bee exesvey suded sce he paper of vo Bahr (97) who have esabshed for he frs e a boud for he reader er CLT; aer o-ufor vara of vo Bahr s resu was obaed by Mrahedov ad abev (976) Vo Bahr s boud has bee proved Coroary of Mrahedov (985) ad Theore of Zhao e a (00) Two ers Edgeworh asypoc expaso resu has bee esabshed by Mrahedov (979) ad Hu e a (007a) whereas expaso wh arbrary uber of ers was obaed by Mrahedov (98) Large devao resus foows fro Theores ad of Mrahedov (996) I hs paper we cosder he geera suao whe he eees of he popuao are rvs whch ay be degeerae (e o-rado) The a of he paper s hreefod Frs Sec we prese a procedure fdg a geera er of Edgeworh asypoc expaso for he sape su; our rue based o sghy geerazed forua of Erdos ad Rey (959) ad used fac ha he egra of ha forua s he characersc fuco (chf) of a depede wodesoa rvecs Ths rue s cosderaby sper ha ha of Mrahedov (98) who have used forua of vo Bahr (97) ad Bozes (000) who gves us wo ers of asypoc expaso for he case whe he eees of a popuao are o-rado ubers Secod we oba Berry-Essee boud ad Edgeworh expaso resu uder weaeed oe codos aey we assue ha EY where 0 s he uber of ers asypoc expaso (oce ha he ora dsrbuo s he frs er) A geera Theores ad of Sec gves he boud for he reader er a uusuay for; hs ogeher wh above eoed fac o he egrad of Erdos ad Rey forua aows o wre esaos of he reader ers er of Lyapuov raos of a assocaed rvs Z see () ad (5) By-ur hs fac aows us o prove cocerg heores of Zhao e a (00) ad Hu e a (007a) by weaeg he oe codos brgg soe correco ad gvg oe ore er of expaso see Rears ad 5 Thrd we oe ha he sape su s a speca case of so aey decoposabe sascs cosdered for sace Mrahedov e a (0) Usg hs fac we derve for he frs e he Ldeberg ype codo of asypoc oray ad Craer s ype arge devao resu These resus are exeso of cocerg heores of Hae (969) ad Robso (977) respecvey Thus he paper copee specru of resus o approxao by ora dsrbuo s coeced Ma asseros are preseed Sec whereas he proofs are gve Seco I wha foows B( p) s used o deoe Berou dsrbuo wh probaby of success p ; sads for he dsrbuo of he rv (rvec) ; c C wh or whou a dex are posve uversa cosas whose vaue ay dffer a each occurrece I he seque oaos we sha suppress he depedece o wheever s covee Procedure of fdg a geera er of asypoca expaso of he chf of S Wre B( p) S Y Y where f Y appears he sape ese 0 wh p / ad codoa o dsrbuo of he rvec rvs ad B( p) Hece S ( ) ( ) S / E e E e E e The o dsrbuo of e ca be vewed as uder : where S S E e / Ee where S Y Y ( ) S e P{ } E e / 0 are depede Ths ogeher wh

3 ad fac ha S ad are a sus of depede rvs pes by Fourer verso q E exp{ S } E exp Y ( p) / q d d( p) q where q p d ( p) : qp qc p q () Y If a are o-rado rea ubers he () ca be wre he for q / q exp{ } exp / d( p) q E S e q p a q d I fac hs s he forua of Erdös ad Réy (959) whch had bee used o prove he frs sasfacory CLT for he sape su The forua had ovaed for a uber of reaed sudes; for sace Hae (960) Bes (969) Babu ad Sgh (985) ad Bozes (000) Se EY E( Y ) ( ) p E Y Z Y x pe Y () x oe ha where ES ad S Z Z Fro forua () we oba S ( ) ( ) : E exp () d( p) (0) q ( ) ( ) d () q Z p ( ) Eexp q because d ( p) (0) due o verso forua for he oca probaby I wha foows us o eep oao spe we pu Z Z (5) oe ha VarZ VarZ ad cov Z 0 (6) I regard o forua () we rear ha he egrad () s he chf of a su of depede wo desoa rvecs Ths fac s cruca our ex cosderaos parcuar cosrucg he ers of asypoc expaso of he chf of he sape su Ideed e EZ ad P ( ) be he we-ow poyoas of ad fro he heory of asypoc expaso of he chf of a su of depede rvecs see Bhaacharya ad

4 Rao (976) p5 (heceforh o be referred o as BR) I our case he cocerg su s Z Z Z where ad / Z Z / p q (7) I s essea ha hs su has zero expecao ucorreaed copoes ad a u covarace arx due o (6) Hece he egrad () beg he chf of he su Z ca be approxaed by a power-seres / coeffces are poyoas of ad coag he coo facor exp{ ( ) / } hece he seres ca be egraed wr over he erva ( ) The degree of P ( ) s ad a degree s ; he coeffces of P ( ) oy vove he cuuas of he rado vecors parcuar: P E Z whose Z of order ess ha or equa o ( ) (8) 6 P ( ) E Z E Z P ( ) (9) Defe fucos () G such ha G0 ( ) ad ad wre G ( ) P ( )exp d / 0 Q ( ) e G ( ) (0) Due o Theore 9 of BR (976) hs seres s ( ) -er asypoc expaso of he egra () Fay we ge he ( ) -er asypoc expaso of () by dvdg Q () wh Q (0) Aso order o esae reader ers hs approxao oe ca use drecy Theore 9 Lea 95 ad Lea of BR a evdey spfed for I parcuar hree er asypoc expaso of he () s / W ( ) e G ( ) ( G ( ) G (0) where e ( ) / EZ 6 6 ( ) EZ 7 ( ) EZ EZ EZ pq ( ) 0 ( ) 6 / e 6 ( ) 6 7 ( ) pq 0 ( ) ()

5 5 p p 0 0 () () E( Y ) E( Y ) () p p 6p p q( p ) () The ers of asypoc expaso of he dsrbuo fuco of S / ca be obaed by sadard way: foray subsue v d u / ( ) ( u) e Hv ( u) / where du ( u) e d u sead of ( ) e / for each he expresso for () / W e see Lea 7 of BR p 5 where H () x s he - v h order Her-Chebshev poyoa Deog () u for hree er asypoc expaso we have u / e H( u) ( u) ( u) wh H () u 6 H5( u) 6 H( u) 6 H( u) pq 0 7 () u H ( u) u H ( x) u u H 5 5 ( x ) u 0 u 5 u Rear Le he eees of he popuao are o-rado rea ubers vz The b ( a a) ( a a) a a a a a a / b A a a u / e H( u)( p) H5( u)( pq) ( u) ( u) A A 6 q Y a Pu () 7q H ( u) ( 6 ) ( ) q Rear If 0 ad 0 he he chf ( ) H u p pq A pq ( ) ca be wre he foowg for pey p ( Y pey ) q ( ) q exp pe exp (5) q q Forua () wh ( ) wre he for (5) has bee used (whou proof) by Zhao e a (00) ad Hu e a (007a) They subsequey proved few eas o he egra of () by worg wh (5) order o ge bouds for he reader ers he frs ad he secod order asypoc of he chf () I coras based o he forua () ad dea ha he egrad of () s he chf of a su of depede wo-desoa rvecs(he fac whch was o reared by he referred auhors) he agorh preseed above s ore srea ad geera

6 6 cosrucg asypoc expaso of ay egh I s aso uch sper ha ha of Mrahedov (98) ad Bozes (000) The forua () assues ha he rvs are asypocay o degeerae hece he case e he case of su of depede rvs s excuded I Zhao e a (00) ad Hu e a (007a) hs ao was covered by usg he approach suggesed by vo Bahr (97) Le ( ) Ee uy 0 ( ) / Y b e Ee / (6) u ( ) B( ) b( ) C r C pc r r r 0 r r vo Bahr (97) have proved he foowg forua ; 0 / pb () e ( ) C 0! (7) I respec of forua (7) we ae he foowg coes Le us wre sead of he C he he rgh- had sde has for exp pb ( ) p B ( ) ; ex usg Tayor expaso dea we ca observe ha he fuco p B () are poyoa ers of where ( )/ / ad are soe agudes depedg o oes of he rvs Y Hece whe we wsh o approxae () C by separae by chf of a ora dsrbuo we shoud approxae fro pb () ad p B () ad ge approprae boud for ( ) p B ( ) p B By hs way vo Bahr (97) derved he boud 60 ax E Y EY / for he reader er CLT; aer Mrahedov ad abev (976) had proved he o-ufor vara of vo Bahr s resu: c ax E Y EY / ( x ) ; whe Zhao e a (00) obaed Berry-Essee boud whch furher prove he resu of vo Bahr see beow Rear 5 Whe we wsh o ge s-ers Edgeworh asypoc expaso addo o ( )/ he above procedure we us separae he ers s propery he su s p Bs p B / s fro pb( ) p B ( ) esae ( ) ( ) ge he asypoc expasos for C ad exp pb ( ) p B ( ) usg Srg s forua ad Tayor expaso dea respecvey Fay we oba he asypoc expaso of () by upyg he asypoc expasos Ths agorh has bee deveoped by Mrahedov (98) gvg he exac forua for he frs four ers; he agorh use ore copcaed agebra copared o ha above preseed based o forua () Recey wo ers Edgeworh expaso wh proved boud for a reader er s gve by Hu e a (007a) I hs paper we cobe boh approaches deveopg he ehods of aforeeoed papers ad assug weaer oe codo s

7 7 Ma resus We s use he oaos of Sec ad ; see ()(5) (7) () ad () Addoay { A} sad for he dcaor fuco of he se A ad sup P S u ( u) ; u where ( u ) ( u ) () u s gve () ad u / ( u ) e ( u) ( u) 6 Theore : Le q The Ldberg codo: : 0 L Z Z for arbrary 0 as () s suffce for 0 Rear The codo () s sasfed f for arbrary 0 ad () E Y pey Y pey 0 qp EY 0 D () where D : p EY are fufed Theore gves weaes codo for asypoc oray of he sape su fro fe popuao of rvs; for he case whe Y are o-rado he codos () ad () cocdes wh he suffce ad ecessary codos of Erdős Rėy (959) ad Hae (960) Aso he codo q s weaer ha ha assued by Hae (960): p s far away fro zero ad oe where Se T / 006 ax ( q) ( ) ( ) EZ () () () () I s easy o see ha E Y pey q () ( )/ () () E Y () ( )/ qp () Theore There exss a cosa C 0 such ha

8 8 ( )/ C ax T q T / T / for ay T T ad each where ad ( ) { } ( ) d d d d d Rears 0 0 d0 d Y ( d0 d) q d exp sup Ee d0 d The 0 0 ( d d ) ( d d ) { d d } (5) Y ( d d ) q d exp q sup Ee d0 d ( d d ) s beer he ( d 0 d ) eees of he popuao are o degeerae rvs I coras ( ab ) uder cera codos such as q s cose o zero bu ay eough of s appcabe for he case whe a eees of a popuao are degeerae e o-rado vz Y a bu q s o oo cose o zero I hs case ( d d ) s expoeay sa for sace f for a gve d0 0 here exs d 0 0 ad 0 o 0 depedg of such ha # : a a / b x for ay fxed x a d d ad eger see Bce ad va Zwe (978) Robso (978) ad 0 Mrahedov (98) / Choosg T ( q) T ad T respecvey we oba fro Theore he foowg: Coroary For ay (0] here exss a posve cosas C where (a) C q / such ha C q 00 / V ( (b) )/ C q 00 / V ( (c) )/ V E Y Tag T ad T see oao () we oba Coroary For ay (0] here exss a posve cosas C such ha

9 9 Se (a) ( )/ ( ) C q 00 / V (b) ( )/ ( ) C q 00 / V p E Y V / ( )/ Theore There exss a cosa C 0 for each ad ad ay such ha for ay (0] C ax T T / T / def ( ) T T 05 Choosg T 05 T ad T respecvey we oba fro Theore he foowg Coroary For ay (0] here exss a posve cosas C such ha (a) C ( ) (b) C 00 / V ( ) (c) C 00 / V Rear We resrced ourseves o us wo ad hree er asypoc expasos oy o eep he eve of copexy of he expressos ow The resus ca be furher exeded for s-er asypoc expaso s bu a he expese of added copexy he proof Rears I s easy o see ha ( p ) everheess Theore ad Coroary gves beer bouds for reader er wr Theore ad Coroary he case whe q cose o zero; parcuar f q = 0 he he cocerg ow resus o su of depede rvs foows fro Coroary Rear 5 The a resus of Zhao e a (00) ad Hu e a (007a) respecvey sae ha C ( q) / C ( q) 00 / V 6 00 / V 6 I coras Par (a) ad (b) of Coroares ad gves ore geera bouds usg he oes of ower order ad 0 respecvey Moreover sce equay () he case do prove of her heores usg Lyapuov s raos sead of where ad respecvey Aso Par (c) gves oe ore er of expaso If Y a are rea o-rado ubers he he a resus of Robso (978) ad Bozes (000) foows fro Coroary

10 0 Rear 6 I ca be ready checed ha Hece he Lyapuov s rao of he rvs 0 p p 0 / EZ (6) approxao by ora dsrbuo coras of Z e s aura characersc esae of he reader ers o par (b) of Coroary ad (6) we coecure ha here exs a cosa 0 Le P ( x) P S x Theore Le pq 0 The for a x 0 x o( ) ad E exph Y ( x) P ( x ) exp x L x O x P ( x ) exp x x L O x ( x) where 0 used by Zhao e a (00) ad Hu e a (007a) Moreover due C such ha C H 0 (7) (8) (9) L ( v) v s a power seres ha for a suffcey arge s aorzed by a power seres wh coeffces o depedg o ad s coverge soe dsc so ha L () suffcey sa vaues of v Parcuary 0 EZ EZ EZ EZ 6 8 v coverges ufory for Coroary Le he codos (7) s sasfed The for a x 0 x o /6 ( ) P ( x) ( x ) O ( x) ad P ( x) ( x ) O ( x) (0) Coroary 5 Le he codo (7) s sasfed The for a x 0 ( ) ( ) ( x) exp x / P x x O x o /6 ( ) () Coroary 6 Le Y a ; ha s he eees of he popuao are o-rado rea ubers If pq 0 he for a x 0 x o( ) he reaos (8) ad (9) hod ad for a 0 reaos (0) ad () hod Aso hs case oao () x x /6 o( ) he

11 p A 6 pq 0 p 6pq A A pq 8pq 8pq Y a x 0 x o / ax a Aso we refer o he recey paper of Hu e a (007b) where was show ha for Rear 7 For he case whe 0 x C q / ax a ( q) / B Theore of Robso (977) shows ha he reao (8) s rue for a /6 / P ( x) ( x) O ( x ) B / q where addo o he oao () we deoe B a a () Rear 8 oce ha he sape su s a speca case of he sascs of for f ( ) so aey decoposabe sascs (DS) a spe rado sape schee whou repacee where f f are fucos defed o he se of o-egave egers s a frequecy of -h eee of he popuao a sape of sze Therefore fro geera heores o DS oe ca derve soe resus for he sape su I parcuar Theores ad are derved fro cocerg heores of Mraedov (996) ad Mrahedov e a (0) For deas of DS see he referred papers ad refereces wh Leas ad Proofs Before geg o proof of he asseros of Sec we sha prove severa auxary eas Acuay Lea ad beow are vad for ay sequece of depede rvecs wh zero expecaos ad ucorreaed sus of copoes; oreover f he rvecs has a fe oes of eger order say he Lea s a coroary of Theore 9 of BR (976); auhors coud o fd eraure cocerg asseros whe he ay be o-eger as we requre I wha foows we s use he oaos of Sec ad ad assue EZ where eger Addoay us o eep he oao spe we pu s o ecessary o be E q p q ( )/ We sha use he foowg reao see forua (0) of Mrahedov (99): for ay copex z eger 0 ad (0] e z z z Re z e () 0! ( ) ( ) here ad everywhere wha foows (0] ad we use he sae sybo for a vaue such ha ahough hey ay o be he sae a a dffere occurrece; aso we use we-ow equaes bewee oes ad Lyapuov s rao: / E E ; We w o refer o hese foruas he cases whe /( ) /( ) s evde fro he coex of oaos We assue ha see he deoes () ad () 0 ad 0 ()

12 he geera case ca be reduced o ha of sasfyg () by cosderg 0 Z Y pey ow o P ( ) Lea If p B( p) 0 Y / sead of Y So fro P ( ) P ( ) P ( ) ( ) ( ) P P c /( ) /( ) ad Se ax () a posve c 0 he here exs C 0 such ha for each 0 ad oe has ( ) e P ( ) C ( )/ e q Proof The case foows fro Par () of Lea A of Mrahedov (005) We resrc ourseves by proof of he case ; he case ca be proved a very sar aer wh ess agebra The proof uses a dea of he proof of Theore 86 of BR p6 We have /( ) /( ) ( ) E Z E Z E Hece we ca wre c () /( ) /( ) r ( ) exp ( ) exp ( ) r oe ha (see (8) ad (9)) P ( ) EZ Z r P ( ) EZ Z E Z EZ Z 6 6 E Z EZ Z To eep he oaos spe we se ow ( ) ( ) ( )

13 Aso where ( ) exp P( ) P( ) exp exp ( ) P( ) P( ) exp ( ) exp ( ) exp A( ) A( ) A ( ) A ( ) A ( ) (5) (6) ( ) exp A exp ( ) ( ) ( ) E Z EZ Z EZ Z EZ Z 6 exp ( ) E Z EZ Z 6 exp ( ) EZ Z E Z EZ Z ex A ( ) A ( ) A ( ) (7) () () () A( ) A( ) ( ) ( ) exp ( ) exp ( ) ( ) ( ) A ( ) A ( ) (8) We have () () ( ) E Z r( ) 6 wh r( ) ad (9) ( ) (0)

14 Usg he as ad frs equaes of () we oba r ( ) ( ) ( c ) r r 6 c c c 8c () ( ) ( ) ow use Tayor expaso dea Eqv (6) he equaes bewee Lyapuov s raos codo () ad Eqv (9) (0) o ge (see oao (8) (9)) ( ) E Z 6 E Z E Z r ( ) 8 P ( ) P ( ) P ( ) r ( ) = E Z 6 r ( ) () () r( ) () ( ) P ( ) r ( ) (5) where ad where Aso r 5 ( ) ( ) (6) r ( ) C( c ) 5 ; (7) r ( ) (8) /( ) /( c ) r ( ) ; (9) 5 ( ) ( ) EZ Z EZ Z E Z EZ Z r6 ( ) (0) 6 r6 ( )

15 5 ( ) ( ) () A as usg () ad ha we have ( ) EZ E Z ( ) ( ) ( ) ( ) c C c () ( ) To prove Lea s suffce o ge approprae upper boud for he odue of A ( ) A ( ) ad her dervave To ge a boud () for he A ( ) : use () wh ad () - (6) ad he as equay of (9); () for he A ( ) : use () wh 0 ad boh equaes of (0) equay () he as equay of (9); () for he A ( ) : successvey o esae A () ( ) use (0) () ad (9); o esae () A ( ) use () wh ad (5) () (8) () (8); o esae () A use () wh ( ) 0 ad () wh he frs equay of (9); (v) for he A ( ) : use o esae A () ( ) er he boud aready obaed for he A ( ) ad (); ex for he () A ( ) er use () () () ad he as equay of (9) Appyg he resus of ()-(v) he Eqvs (5) (6) (7) ad (8) aso usg he equaes bewee Lypuov s raos he codo () wh soe agebra oe ca copee he proof of Lea ; he deas are oed Lea Le q has / ax 00 If 05 ad 05 q he for each 0 oe ( ) exp 0 Proof of Lea foows fro Par () of Lea A of Mrahedov (005) Lea There exss a 0 each 0 oe has / c such ha f c q ( ) exp009q ( ) ad 0065 q q he for Proof For a gve rv say e * where s a depede copy of We have

16 6 Z Z ( ) (0 ) E e e E e (0 ) E Z EZ Usg hs ad he equaes xexp{ x } we fd for = 0 ( ) ( ) ( ) ( ) ( ) e exp (0 ) () O he oher had E exp{ } exp pqs / c u e s e u 0 u c c Hece for 0065 q q we have (0 ) ( ) / e q Ths equay ogeher wh () yed e ( ) exp e q q e exp q e ( c ) c exp 009q / sce foows c ( q) c q ad here we choose c ( e ) /6( ) 006 Lea There exss a c 0 such ha f 0 oe has c ( ) exp cq Lea see () ad 0065 q q he for each Proof The proof foows by usg he ewo-lebz rue for dervave of a produc reasos very sar o ha of he proof of Lea of Zhao e a (00) ad equay Here 0 Le ( ) () c so sa ha c c 8 cos(/6) / hece 0c 009 ( ) exp / W / ( ) W ( ) e 6 ad W () be defed () Reca he oao () () ad oe ha (5)

17 / 00ax ( q) T Lea 5 For each ad 0 oe has: f he here exs a cosa C 0 such ha / T T 00ax ( q) q / ( ) W( ) C e e Proof We assue ha ess agebra Le We have / ax 00 cq T see oao of Lea ; he case T Q e P d ( ) ( ) ( ) ( ) ad 7 T s sper eedg cosderaby 05 q ( ) d 05 q q ( ) d / / e e P ( ) d D D D D (6) Appyg Lea ad o esae D D ad D respecvey afer spe cacuaos we oba I s ready see ha / cq D D D C e e (7) ow e / q/ D Ce e T Wre (8) ( ) Q ( ) ( ) d D Q ( ) Due o defo of () 05 q D D Q () (9) Q easy o see ha: for T / Q ( ) C e (0) oe ha T 00 T T 00 hece we ca use Lea o esae D ; aso o esae D we appy Leas ad ; fay we oba: f T he

18 8 / cq () D D C e e Thus fro (6)-() we have: f T he for 0 ad / ( ) Q ( ) C e e O he oher had usg Srg s forua s easy o show ha cq () (0) G (0) () ( q) Appyg () ad () he forua () ad usg equay bewee Lyapuov s raos wh que evde cacuaos we copee he proof of Lea 5 Rear I ay reared ha Lea 5 ca be furher exeded for s-er asypoc expaso ad ay 0 s s bu a he expese of added copexy he proof Such a exeso of Lea 5 ca be used o derve asypoc expaso of he oes of he sape su I parcuar oe ca show ha see oao () / ( )/ ( ) E S ES C q pq ( /) 0 E S ES C ( q) Y I he case () a are o-rado rea ubers hese foruas have he foowg for see oaos () ad ( ) ( ) E S ES p A C q pq A 6 p ES ES ( q) ( 6 pq) A C ( p q )( B5 ) q ( q) A upper boud for Lea 6 Le The for c ( ) / ES has bee preseed Lea 6 of Rose (967) ( q) ( )/ / () / c ( ): ( )( ) /6 oe has Proof Se ( ) exp p v s Sce s E Y vs ax( vs ) / ( ) VarY s s V ad () we have q V q qv q V q hece /( ) / /( ) V v /( ) /( ) Therefore q 0 / / (5)

19 9 Appy () wh = o ge Ee Y / v c( ) c( ) / ( )( ) ( ) Usg hs equay ad ha Y / q pq Ee p we oba Y ( ) E exp q p Ee pq Y / Hece for Y / v p Ee p c( ) p c ( ) we have ( ) exp p c( ) p exp p c( ) c ( ) sce (5) ad ha p exp p exp Lea 6 s proved Lea 7 Le p / ad The for ( ) / W ( ) c e where W() are defed () ad (5) 0 oe has /( ) Proof Proof based o he forua (7) of Bahr hece we use deoes (6) Aso we sha deveop soe deas of Mrahedov (98) Zhao e a (00) ad Hu e a (007a) Le c ad / 0 ( ) / ( ) ( ) ad equaes bewee oes we observe ha c ( ) 0 c ( )( )! 0 wh Due o () 0 Hece V (6) / V (7) ( )/ ( )/ If c ( ) (8) /( ) he aso ad ( ) c ( )/( ) (9) / c ( ) / 0005 (0) /( )

20 0 d EY s Se s Appyg () ad (6)-(9) wh approprae ad respecvey we oba ( ) b d d d d 6( ) ( ) ( 0 ) 0 v () ( ) d 6d 0 0 ( ) 6 ( )/( ) d ( ) v b ( ) ( 0 d) d 0 d () 6( ) d v ( ) ( 0 ) b d () b () d v () v b ( ) (5) Le us prove he case oe ha /( ) v / 0 (6) Usg (7)(8) (6) equay v v v ad successvey ()-(5) we oba /( ) For ad ( ) ( ) (7) 6 6 ( ) pb p ( ) p B ( p) p 0 p ( ) () p (8) ( ) ( ) p B ( p) p 0 p 0 0 (9) ( ) p B ( p) p 0 (50) /( ) 5 p B ( ) 0 05 (00) 0 /( ) for 5 (5) fro (7)-(5) we have p B( ) (5) 5

21 ( ) ( ) p B( p) p 6 0 (5) p (5) 0 8 sce ad hece Thus (5) ad (5) gves /( ) /( ) / p B ( p) p 0 (55) ex we sha use argues sar as he reaos ()-() of Mrahedov (98) wh appcao above Eqvs (6)-(55) Use he Srg s forua o ge for 0 r p p 0 ( ) ( ) C r r r r (56) ow rewre (6) he for where 0 / () e I I I (57) pb () I C! here he suao s over a 0 ad 0 for a eas oe 5 ; pb () p I C 0! pb () p 0! I Because C r foows fro (5) ad (55) exp ( ) exp ( ) exp 0 5 (58) I p B p B c p Usg (56) we oba pb () 0! I c exp ( ) ( ) ( ) p B p B p B As Mrahedov (98 equay (7)) we have c 6 ( )exp p 0 (59)

22 I p B( ) q p B ( ) ( ( ) ) p B( )!!! 0 0 p B () ( ( ) )! 0 q q exp p B ( ) p B ( ) p B ( ) c p B ( ) p B( ) where 0 ad c are soe cosas (we do o eed o have c expc for ahough ca be foud easy as a og forua) Use (7)-(50)(5) (5) equaes q 0 q 0 afer soe agebra we oba for 0 /( ) I ( ) ( ) exp p 0 6 ( ) pq( ) 0 qp 0 c (60) The case of Lea 7 foows fro (57) - (60) Proof of he cases ad s a very sar o proof of he case wh que evde agebra usg Eqvs (7)-(0) wh respecvey ad equay are oed 0 ( ) C r r sead of (56) The deas Proof of Theore foows (see Rear 7) edae aer fro Theore of Mrahedov e a (0) by f () x Y x pug ad B( p) Proof of Theore By Essee s soohg ea ad he fac ha we have ( ) W( ) ax ( u) W( u) u u ( ) W( ) d (6) T T ax ( u ) W ( u ) u u ( ) W( ) W( ) + d d T T T () d J J J J (6) T T T T W are defed () ad (5) ad where () T fro Lea 5

23 Usg Lea 5 ad ag o accou he expoea facor of W () we oba J J J C ( q) ( )/ Theore foows fro (6) ad (6) because J T / T / Y ( ) I vew of equay Y Ee p Ee Y Y Ee p Ee Therefore ad we have (6) Y ad Ee pq E cos( Y ) Ee exp Ee Y Y (6) Y Ee exp pq E cos( Y ) O he oher had by forua () Y exp q Ee (65) q Y / / (66) T / T / T T Ee d d (0) The equay (5) foows fro (6) (65) ad (66) Proof of Theore Le p / he q / ad ag o accou ha ad (8) we have ( / / / )/ ( )/ ( )/ ( )/ ( ) q V V herefore ( )/ ( q) / ( ) sce ( )/ Thus f p / or Le ow p / ad where ( ) ( )/ / q ( ) ( p ) (67) he Theore foows fro Theore ( )/ / q ( ) W( ) W( ) d d T T T ad T /( ) 0 I vew of (6) we have ( ) ( ) d d T T T T T T c ( ) c ( ) s a cosa defed Lea 5 Appyg here Lea 5 6 ad defos of W () afer spe cacuaos we copee he proof of Theore Proof of Coroary oe ha

24 00 ( ) T ad T T / V / q (68) / Therefore Coroary foows edaey fro Theore by pug T 00 ( q) T ad T for he cases ()() ad () respecvey Proof of Coroary Le 0 / The p p p p p p p p p / / Use hs fac ad equay bewee oes o ge for Hece () ( )/ E Y pey ( / 0) / p ( )/ Le 0 / The ( )/ / ( ) / () ( )/ ( )/ (69) () ( )/ ( ) / 0 hece () ( )/ ( )/ / (70) () () Pu ow Theore T use (68) ad (69) (70) wh ad respecvey o copee he proof of Coroary Proof of Coroary Par (a) edaey foows fro Theore by pug ax () () equaes (68) (69) ad (70) wh T 05 Therefore Pars (b) ad (c) foows fro Theore by pug ad respecvey Proof of Theore foows (see Rear 7) fro Theore of Mrahedov (996) by pug f () x Y x B( p) ad og ha T oe ha ad usg q s bouded away fro zero Proofs of Coroares ad 5 foow fro Theore edaey because we ca assue whou osg of geeray ha ax Referece a c Babu G J ad Sgh K (985) Edgeworh expaso for sapg whou repacee fro fe popuao J Muvarae Aa Babu G J ad Ba Z D (996) Mxures of goba ad oca Edgeworh expasos ad her appcaos J Muvarae Aa vo Bahr B (97) O sapg fro a fe se of depede rado varabes ZWahrsch Verw Geb Bhaacharya R ad Raga Rao R (976) ora approxao ad asypoc expasos Wey ew Yor

25 5 Bce P J ad va Zwe W R (978) Asypoc expasos for he power of dsrbuo-free ess he wo-sape probe A Sas Bes A (969) O he esao of he reader er he cera heore for sapes fro fe popuaos Suda Sc Mah Hugar 5-5 (Russa) Bozes M (000) Oe ad wo-er Edgeworh expaso for fe popuao sape ea Exac resus I;II Lh Mah J 0-7; 9-0 Cochra WG (96) Sapg Techques Wey ew Yor Erdös P ad Rey A (959) O he cera heore for sapes fro a fe popuao Fub Mah Is Hugara Acad Sc 9-6 Hae J (96) Lg dsrbuos spe rado sapg fro a fe popuao PubMahIs Hugar Acad Sc Hos L (979) A ufed approach o heores for ur odes J App Probab65-6 Högud T(978) Sapg fro a fe popuao A reader er esae ScadJ Sasc Hu Z Robso J ad WagQ(007a) Edgeworh expaso for a sape su fro a fe se of depede rado varabes Eecroca JProbab 0-7 Hu Z Robso J ad WagQ(007b) Craer-ype arge devaos for sapes fro a fe popuao A Sas Mrahedov Sh A ad abev I (976) O esaos of he reader ers he heores for he sape sus fro fe popuaos of he rado varabes L heores ad Maheaca Sascs 0-(Russa) Mrahedov ShA (979) Asypoc expaso of he dsrbuo of sape su fro fe popuao of he depede rado varabes Repors of Uzbe Acadey of Sceces (Russa) Mrahedov Sh A (98) A asypoc expaso for a sape su fro a fe sapetheory Probab App Mrahedov ShA (985) Esaos of he coseess of he dsrbuo of decoposabe sascs he uoa schee Theory Probab& App [Mrahedov SM Jaaaadaa SR ad Ibrag BM (0) O Edgeworh expaso geerazed ur ode J Theor Probab DOI 0007/s z Mrahedov SA(996) L heores o decoposabe sascs a geerazed aocao schees Dscree Mah App Robso J (977) Large devao probabes for sapes fro a fe popuao A Probab Robso J (978) A asypoc expaso for sapes fro a fe popuao A Sas Rose B (967) O he cera heore for a cass of sapg procedures Z Wahrsch Verw Geb Wad A ad Wofowz J (9) Sasca ess based o peruaos of he observaos A Mah Sa Zhao LC Wu C Q ad Wag Q (00) Berry-Essee boud for a sape su fro a fe se of depede rado varabes J Theor Probab