On nonnegative integer-valued Lévy processes and applications in probabilistic number theory and inventory policies

Transcription

1 Amrca Joural of Thorcal ad Appld Sascs 3; (5: - Publshd ol Augus 3 3 (hp://wwwsccpublshggroupcom/j/ajas do: 648/jajas35 O ogav gr-valud Lévy procsss ad applcaos probablsc umbr hory ad vory polcs Humg Zhag * Jao H Hal Huag Dp of Mahmacs ad Sascs Cral Cha Normal Uvrsy Wuha Cha Emal addrss: a @gmalcom(zhag H To c hs arcl: Humg Zhag Jao H Hal Huag O Nogav Igr-Valud Lévy Procsss ad Applcaos Probablsc Numbr Thory ad Ivory Polcs Amrca Joural of Thorcal ad Appld Sascs Vol No 5 3 pp - do: 648/jajas35 Absrac: Dscr compoud Posso procsss (amly ogav gr-valud Lévy procsss hav h propry ha mor ha o v occurs a small ough m rval Ths sochasc procsss produc h dscr compoud Posso dsrbuos I hs arcl w roduc approachs o prov h probably mass fuco of dscr compoud Posso dsrbuos ad w oba sv approachs o prov h probably mass fuco of Posso dsrbuos Fally w dscuss h coco bw addv fucos probablsc umbr hory ad dscr compoud Posso dsrbuos ad gv a umrcal ampl Surg Posso dsrbuos (a spcal cas of dscr compoud Posso dsrbuos ar appld o umrcal soluo of opmal (s S vory polcs by usg couous appromao mhod Kywords: Probably Mass Fuco Nogav Igr-Valud Lévy Procsss Probablsc Numbr Thory Dscr Compoud Posso Dsrbuo (S S Ivory Polcs Iroduco Posso dsrbuo s a famous dsrbuo dscr dsrbuo famly ad has mpora applcaos socal ad coomc sccs physcs bology ad ohr flds For ampl h umbr of passgrs cam o a bus sop h umbr of parcls md by radoacv subsacs h umbr of mcroorgasms a rgo udr h mcroscop ad so o Posso (837 [5] rd o us h bomal dsrbuo of svral prms o drv dsrbuo fuco of h Posso dsrbuo hs rprsav work Rchrchs sur la probablé ds jugms maèr crmll maèr cvl précédés ds règls géérals du calcul ds probablés 3 ω ω ω P ( + ω Afr abou yars d F (99 [6] ad ha h probably of a v h rval [ + of Posso Procsss s λ + o( ( λ > ad h probably mor ha o vs s o( Khch [] summarzd h quval codos of Posso ω dsrbuo: zro al codos saoary crms dpd crms ad ordrlss (mpossbl of wo or mor vs occurrg h sam mom of m I acual lf Posso procsss (dsrbuo s o a adqua modl for h obsrvd daa ha h possbly of wo or mor vs occur a a gv sa Khch [] also gralzd h Posso procsss by gvg mor rsrc of ordrlss As for a suao: durg prod of h umbr of cars ha arrv a h rmal sao ca b rgard o Posso dsrbuo vry car ak passgrs wh probably α hus h umbr of passgrs who arrv a h rmal sao ca o sasfy Posso procsss So w assum ha hr ar may vs occurrg a small sgm of lgh Hgh [4] cosdrd aohr suao: sombody mgh hrow som lrs (mor ha o o a posbo a h sam m Dscr Compoud Posso Modl Nogav Igr-Valud Lévy Procsss W mploy h followg dfo du o [4] Dfo (Dscr compoud Posso procsss

2 Amrca Joural of Thorcal ad Appld Sascs 3; (5: - Nogav gr-valud sochasc procsss { X ( } sasfy h followg four codos: ( Ial codo: X ( ; ( Saoary crms: h vs occur [ + oly dpds o ad s o rlva o ; ( Idpd crms: h vs occur [ + s dpd wh h vs whch happ bfor ; (v Suprmposo: h probably of vs akg plac bw [ + s: P( λα + o( ( λ > α α Espcally wh appls o h vory maagm hory h umbr of cosumrs comg a prod of ca b s o subjc o h surg Posso dsrbuo (SPD Wh h rval s small ough ad h umbr has h propry of gomrc dsrbuo ha s a ( α α ( Gallhr [] appld SPD o h vory maagm frsly ad amd as surg Posso dsrbuo W wll prov h plc prsso of probably mass fuco (pmf of SPD h followg par Dfo was pu forward by Khch [] whr α b probably of ogav dscr dsrbuo Hgh [4] sll amd surg Posso dsrbuo sad of dscr compoud Posso dsrbuo sg h broad ss of vory maagm Thr ar som ohr ams: Pollaczk-Grgr dsrbuo Gralzd Posso dsrbuo Composd Posso dsrbuo Posso powr srs dsrbuo Posso par grapps dsrbuo Posso-soppd sum dsrbuo Mulpl Posso dsrbuo If dvsbl dsrbuo o h ogav gr Radrs ca fd mor gral formao of Posso dsrbuo ad dscr compoud Posso dsrbuo [8] ad [4] Th wd rag of applcaos wh dscr compoud Posso s [] [6] [34] [] ad [] Dfo (Dscr compoud Posso dsrbuo I parcular w say h dscr radom varabl X sasfyg h codos abov has a dscr compoud Posso (DCP dsrbuo wh paramrs ( α λ α λ R ( α λ > α W do as X CP( α λ α λ If X CP( αλ αrλ w say X has a DCP dsrbuo of ordr r Rmark I h Sco 38 w wll show X ( hav compoud Posso sum proprs Thorm Dscr compoud Posso procss X ( s a Markov cha wh saoary dpd crms L P( P { X ( X ( } Th pmf of DCP procsss s P ( Σ k k N k k k α α α k + k k ( λ k k k ( whr N { } By h dfo of Bll polyomal [4]: hc P ( sasfs a B ( a a p[ ] B ( α λ α λ P( So P ( ca b prssd by Bll polyomal Th drma quao Σ k ( k N s calld Dopha's quaos Th dph horcal proprs ad sascal applcaos of Bll polyomal wh Dopha's quaos Σ k s [3] Thorm If X ( CP( αλ αλ h h probably grag fuco (pgf of X ( s λ α ( s ( P s ( s ( Rmark W wll gv approachs o proof h pmf of DCP dsrbuo ad wo o approachs proof h pgf of DCP dsrbuo Sco 3 Lévy procsss { X ( } sasfy h followg four codos as follows: ( P{ X ( } as ; ( X ( has dpd crm; ( X ( has saoary crm; (v X ( s almos surly rgh couous wh lf lms Lévy procsss ca drv Lévy Khch formula (dals ca b s Bro(996 Th characrsc fuco X ( E[ ] θ of X ( s p aθ σ θ + ( θ θ I \{} X w( d R < whr a R σ I A s a dcaor fuco Ad w( d s a masur whch s calld Lévy masur I sasfs < R\{} m{ w( d Lévy Khch formula s frsly obad h gralzao of d F [6] ad Kolmogorov [] rsuls by Lévy [3] ad Khch [] Ths formula ca b prssd as h uo of Brow moo cosa drf

3 Zhag Humg al: O Nogav Igr-Valud Lévy Procsss ad Applcaos Probablsc Numbr Thory ad Ivory Polcs compoud Posso procss ad a pur jump margal By h dfo of Lévy procss suppos w( d αλdδ (3 whr δ s Drac masurs (or po masur wh fdδ f ( ( N ad < R\{} m{ w( d Bcaus X ( s a gr valu as wll as h dfo of w( d w hav θ σ θ ( θ I < ( a R\{} X w d Hc h characrsc fuco of X ( s θx ( jθ E[ ] p[ λαj ( ] j Thorm shows ha h Lévy masur (3 w dfd bfor s rasoabl Dscr compoud Posso procss s a couous-m o-gav gr valu Lévy procss Bardorff [] pu forward h gr-valud Lévy procss ad h us lacy facal coomrcs Spcal Cas Hrm Dsrbuo Wh r dfo calld Hrm dsrbuo Th pmf s P ( [ ] α α( λ ( Th pmf abov s go by h pgf of Hrm dsrbuo padd rms of Hrm polyomal Mor dals ca b s Kmp [9] Surg Posso Dsrbuo I vory sysm Gallhr [] us h DCP dsrbuo wh paramr X ( CP( αλ αλ o dscrb h dmads whr αλ ( α α λ H calld surg Posso dsrbuo Th followg pars show ha h pmf of SPD ca b prssd by Lagurr polyomal W d a lmma Th powr srs soluo of scod-ordr lar dffral quao y + ( y + y d ( sy L ( whch s calld Lagurr d polyomal W bg by provg a propry of Lagurr polyomal Frs w d Lmma I s h coco bw Lagurr polyomal ad pgf of SPD Lmma : Th Taylor s formula wh rspc o of P( s L ( r < d ( Proof: I s asy o prov ha sasfs d ODE: y + ( y + y Suppos s a compl varabl ad assum ha ( P( ( s aalyc r < Df h Taylor paso of P( r < s a ( By h Cauchy formula of hghr ordr drvav of aalyc fuco h coffc a( of powr srs s ξ z ξ z dξ ξ r + r + ξ z π ( ξ π ( z Thorm 3:Th pmf of SPD s d ( d α α P ( α [ L ( λ L ( λ] λ α α Proof: Th pgf of SPD ca b rwr as follow by h Lmma abov: α αs ( α α λ( s λ α αs P( s α L ( λ ( as ( as α α α λ + [ L λ ( as L λ ( as ] α α P ( s h coffc of s h padd formula of P( s hc P ( P( s α α α [ L ( λ L ( λ] α α s L α 5 λ plo h graphc SPD ad PD of y P ( by Mapl 6 Fgur α dz

4 Amrca Joural of Thorcal ad Appld Sascs 3; (5: - 3 c + c c ( c c c ar cosas (4 m m Lmma 4 (Polyomal of -h powr m ( α m α + m m k k k α α α l + ( m + k k k k + k + k m k N k k k l (5 Fgur α 5 Th fgur of SPD s mor dwarfd ha s PD from Fg ad Fg Ths s bcaus h SPD hav suprmposd vs occurrd wh a suffcly small m rval whl h PD s o allowd Numbr of cds a u m s rlavly mor SPD Mor dscr compoud Posso dsrbuo ampls ca b s Wmmr [3] lss mor ha spcal cass 3 T Approachs o Prov h PMF of Dscr Compoud Posso Dsrbuo 3 Lmma Lmma (Cauchy fucoal quao Suppos f s couous o R so f ( + y f ( + f ( y for all y R Th f ( a( a R Th proposo of Cauchy fucoal quao: Suppos f s couous o R so f ( + y f ( f ( y for all y R Th a f ( ( a R Lmma 3 (Eulr s mhod of lar dffral quao wh cosa coffcs Suppos a p-dmsoal colum vcor fuco P s a -ordr coffc mar of dffral quao d d P L F( λ P λe ad d F ( λ s h characrsc quao of P d F( λ has dffr characrsc roos h compl fld Suppos hy ar λ λ λ m( m whch hav algbrac mulplcs r r r ( r + r r m Th h parcular soluo of h quao s [ E + F( λ + F( λ + r r λ + F ( λ ] A ( m ( r whr h vcor r A sasfs quao F ( λ A Hc h gral soluo of quao d d P s m Aohr prsso ca b s by Thorm [3] I l ordr o smplfy h symbols s h coffc of as N l m k k k α α α m k k k k + k + k m k N k k k l Lmma 5 (Nlpo mar I s a -dmso I dy mar l shf mar b N f + h N Lmma 6 (Faà d Bruo formula [] If g ad f ar fucos wh a suffc umbr of drvavs h w hav d g[ f ( ] [ k k d k + k k kj k + k k f ( f ( f ( ( g f ( ( ( k k k ( ( ( ] I s asy o vrfy Lmma o Lmma 5 ad h proofs ca b foud may books 3 Uvara Mulomal Dsrbuo Appromao Now w dscuss h gralzao of ms of Broull rals ad cosdr udr h suao: Ths cas s smlar o uvara mulomal dsrbuo (s [8] p5: Suppos ha hr s a squc of dpd rals durg sgm lgh of whr ach ral has f possbl oucoms A A A ha ar muually clusv ad h probably of ach oucoms ca b wr as : p λ + o( ( p α p p( A λα + o( rspcvly L h occurrc of A ( b dmd o b quval o succsss ad h occurrc of A b dmd o b a falur Th umbr of succsss X ( achvd h rals Wh + h radr ca s Fg3 w dvd [ + o N pcs vly ad vry rval s ha s lm N S hm N

5 4 Zhag Humg al: O Nogav Igr-Valud Lévy Procsss ad Applcaos Probablsc Numbr Thory ad Ivory Polcs [ + [ + + [ + ( N + N Thr s oly o vs choosg from A A A vry [ + h + ( h + ( h Th umbr of vs A A happd all small rvals ar lmd whl h umbr of A happd durg [ + s f To udrsad h procss asr w mag hr s a dc wh f surfacs ad ach surfac wr o umbr Durg m rval [ + afr N (approma o a gr ms of ossg fgur ou h probably ha oal umbr s Fgur Dvd o fy may rval of uformly Wh N + DCP dsrbuo s qual o a codoal mulomal dsrbuo ha was producd by a gralzd f ms dpd rpo ral h pgf o ral s + G( s ps ( s pgf P( s [ G( s] N of DCP ca b wr as: k + k + k N ku N k + k k Th k k k N p p p lm[ p ( N s + ] N k k k by Lmma 5 Accordg o h pgf w kow P ( s h coffc of s h paso formula of P( s sc αλ λ p ( p N αλ N N whr k k ar fd ogav grs ad h cosra s u ku N Th P ( lm N p p k k k N k + k + kr N ku N k + k k ( α λ α λ N α λ α λ lm k k k k lm[ N ( k k ] N k k ( ( N k + k + k N ku N k + k k N( N [ N ( k k + ] lm N k k N k k ( αλ ( αλ k k k + k k k u N 33 Sysm of Dffral Equao p k k k k k Nw w cosdr X ( CP( αλ αλ If X ( CP( αλ αrλ wh > r w hav α If w hav P ( + P ( P ( By Chapma-Kolmogorov quao ( also calld h oal probably formula solvd h quao abov by Lmma h λ P ( λ + o( ( λ > By h dpd crm Chapma-Kolmogorov quao ad suprmposo s asy o s ha P( + [ λ + o( ] P( + [ λα + o( ] P ( [ λα + o( ] P ( P ( + P ( λ[ P ( + αp ( α P ( + α P ( ] + o( L w oba a dffrc-dffral quao wh al codos: P ( λ λ[ P( + α P ( + + α P( + αp ( ] P ( (6 Equao (6 ca also b rprsd by mar as follow: P ( λ λα λα P ( P ( λα P ( P ( λ P ( W ca oba h quao P ( QP ( ad h characrsc quao of h coffc mar P s d( P λ E ( λ λ + so λ s a characrsc + roo of P Th soluo of F ( λ A s a arbrary cosa vcor Furhrmor ca s a cra vcor by h uquss of pmf ad (4 ad wo chagd wh h chag of α α Cosdrg a rm cas: f α h w hav whr * ( λ T P ( ( λ P * ( ca b prssd as I + + ( λ ] ca * I ( λ ( [ E+ λ P By Lmma 3 ad I s a -dmso dy mar Thus w oba

6 Amrca Joural of Thorcal ad Appld Sascs 3; (5: - 5 ( λ ( λ λ ca λ λ Solv hs sysm of quao w oba c A ( T Usg h prsso of parcular soluo o solv F( λ ca F ( λ ca w ca g α α α α α ( λ c λ α F A α λ λ( α N + α N α N ca λ( N N N N F ( λ ca λ N ( ( α N + α N α N ca λ N N F A N N + + N ca ( λ c λ ( α + α α T λ ( N N ( λ c λ ( α + α α λ ( N T F A N N + + N ca Accordg o Lmma 3 wll b ocd ha h uqu soluo of (6 s as follow: ( P ( P ( P ( T T T T ( λ [( + ( N N λ + ( N N T ( λ T ( λ ( N N ( N ] By h dfo of h frs row of P ( 34 Mar Dffral Equao T T λ N o show ( w oly cosdr Accordg o h proof sysm of dffral quao mhod w hav h dy mar form hus P ( QP ( m Q ( I + αn + αn αn m P ( C ( λ C m m L hc w hav h al codo: T Q ( lm P ( lm C C By Lmma 4 w hav m ( I + αn m ( I m m k k k m m ( E α α l αn ( N k k k m k + k k m k N k k k l P ( [ k l k + k k m k N k k k l k k α α α k k m k k k l ( ] λ ( ( λ k N C k k k k k α α α ( λ k k k l m k + k + k m k N k k k l N C k k l Bcaus of N l C ( T f w choos h frs row of P ( ( s asy o b prov by h sam abov Th followg wo mhods s h drvao of h pmf by pgf 35 Faà d Bruo Formula Frsly w gv wo approachs o prov Thorm Smlar o h bomal dsrbuo approma Posso dsrbuo as wll as h buldg mhod h Sco 34 s asly o oba h paramr lmd codos of mulomal dsrbuo approma o DCP by pgf: P( s lm [( p + ps ] Npk αλ N λ N lm[ + ( α] N N αs N λα ( s Or by dfg h pgf asp ( s P ( s subsu o (5 w ca g h prsso as a ODE Thus w hav I Q( s [ ( λα s ] Q ( s Q( s c λα ( s L s by h dfo of pgf w kow ha ( s h soluo of h ODE abov Th accordg o h vrso formula of grag fuco as wll as h gralzd form of Lbz formula Faà d Bruo formula s mmdaly o g h

7 6 Zhag Humg al: O Nogav Igr-Valud Lévy Procsss ad Applcaos Probablsc Numbr Thory ad Ivory Polcs pmf of DCP By h pgf of DCP w hav {( j [ ( ] ( j } kj ( kj αλ s α s λ ( s ( [ λ α ] λ s d λ α ( s P ( [ ] s ds k k k α α α [ ( λ ] k k k k + k k ku N k + k k L rug from o ad h Thorm s provd 36 Cauchy S Igral Formula Ulzg h rlaoshp of Cauchy formula of hghr ordr drvav ad powr srs as wll as formula (4 w hav h pmf + α λz P ( dz C + π z πc + z m m + k l k + k k m k N k k k l k k + ( ( λ [ π C z k k k α α ( λ k k ( + α z + α z l ] zdz m m ( λ dz k k α α ( λ l ( zdz π C z k k k k + l k + k k m k N k k k k k l By h rsuls of compl grao a s a ral po C : dz π C ( z a Z Wh l ( s clarly 37 Fucoal Equaos Wh h Chapma-Kolmogorov quao maks obvously ha P ( + P ( P ( By Lmma h w hav λ P ( λ + o( ( λ > (7 Wh smlarly w hav: P( + P ( P ( + P ( P ( (8 By Lmma P ( α λ ( α s clarly o s by (7 ad (8 Coually w hav formulas blow: P ( [ αλ + α ( λ ] λ ( α P ( Σ k k N k k k α α α ( λ k k k k + k k Th w us mahmacal duco mhod o prov ha ( s ru Morovr w hav P( λα + o( ( ad s asy o s h paramrs of DCP procss α mus b ogav cosa du o P( 38 Compoud Posso Sum DCP procss X ( ca b dcomposd as X ( Y + Y Y N( whr Y s d ogav gr-valud radom varabls wh P{ Y j} αj ad N( P( λ W frs vrfy ha h pgf s formula ( by h codoal pcao ad dpdc: X ( X ( Es E( s N( P( N( Y (E s λα ( s ( λ Hc by dpdc w ca fgur ha N ( P ( P{ Y } ( α λ P{ k Σ k k N} k αλ k Σ k k N k k k α α α ( λ k k k 39 Sum of Wghd Posso k + k k DCP procss X ( ca b dcomposd as X ( Z ( + Z ( Z ( + whr Z( s dpd of ach ohr ad Z ( P( λα W frs vrfy ha h pgf of X ( s formula (3 by h codoal pcao ad dpdc: X ( Z ( Z ( Es Es E( s λα (s λα ( s To vrfy s pmf by dpdc w hav:

8 Amrca Joural of Thorcal ad Appld Sascs 3; (5: - 7 P ( P{ Z ( } ( α λ P{ k Σ k k N} k αλ k Σ k k N α α k k k k k α k k + k k ( λ Rmark 3 Th pcao of DCP procss X ( s E X ( E[ Σ Z ] Σ α λ E X ( s f ff Σ α < By dpd crms propry E{ X ( X ( s + X ( s X ( τ τ s} + E{ X ( s X ( τ τ s} X ( s X ( Σ α λ sasfs h dfo of couousm margal E( X ( < ; E( X ( { X ( τ s} X ( s( s Hc X ( Σ α λ s calld dscr compoud Posso margal Th varac of dscr compoud Posso procss X ( s D X ( D[ Σ Z ] Σ α λ D X ( s f ff Σ α < Hc X ( s a squar-grabl margal Dfo 3 (Squar-grabl margal[7] p59 A squar-grabl margal { M( } such ha ( τ τ E [M( M ( s] { M( s} s ( s s calld a ormal margal Th compsad ad rscald procss X ( Σ αλ M( Σ α λ s a ormal margal (S Chapr of [7] 3 Rcursv Formula W bg by dfg h dcaor fuco j gj( j j Th k R by h compoud Posso sum Sco 38 follows ha + P ( P{ X ( } g ( P( X ( E[ X ( g ( X ( ] ad N( k ( λ E[( Y g ( X ( ] ke[ Yg ( X ( ] k k k ( λ jα λe g ( Y + jy j ( k j k k j j j k j k P j( k ( λ λ jα λ jα jp j( ( k N ( P ( P{ Y } P{ N( } λ By usg h rcursv formula ad usg mahmacal duco mhod s clarly o g quao (6 3 Covoluo Accordg mulomal dsrbuo paso formula compoud Posso sum propry ad Lmma 4 w g P{ X ( } P{ X ( N( } P{ N( } whr k ( λ P{ Yk } k ( λ P + + s { ( ps ps } k k ku N k uku ++ k k k p p k k ( λ λ k k P{ Yk } s pmf of h covoluo of Y k 3 Rmark of h Mhods Argug from h Chapma-Kolmogorov quao Whakr [33] obad h dffrc dffral quao rlag o P ( solvg o by o h gav h frs 5 probably prssos of P ( ( 34 Jaossy [7] drcly solv h Chapma-Kolmogorov quao o g h prsso of P ( shows ha h suprmposo codo h dfo of DCP procss ar ucssary Ludrs[4] am Pollaczk-Grgr dsrbuo mmory of hs wo popl's fdg ad drvd by h mhod of sum of wgh Posso Hofma [6] drv h prsso of P ( by h us of Faà d Bruo formula from h pdf of P ( Adlso [] dffra pgf of DCP dsrbuo for svral ms ad obad h rcurrc rlao of pmf Dscr compoud Posso dsrbuo s mos rsg dscr dsrbuo s hard o fd ohr dsrbuos hav such may proprs ad h mulpl approachs of provg h probably mass fuco of

9 8 Zhag Humg al: O Nogav Igr-Valud Lévy Procsss ad Applcaos Probablsc Numbr Thory ad Ivory Polcs dscr compoud Posso dsrbuo Amog approachs h sysm of dffral quao covoluo Mar dffral quao Cauchy s gral formula ar orgal works h ohr mhods gv a dal ad vvd proof from ohr works L α DCP dsrbuo cp for mhod of compoud Posso sum sum of wgh Posso ad covoluo w oba sv approachs o prov h pmf of Posso dsrbuo Wh coms o paramr smao of DCP s [8] ad [34] 4 Probablsc Numbr Thory I h probablsc umbr hory prm dvsors of grs ad h dsrbuo of prms shor rvals hav proprs of Posso dsrbuo udr som codo Th dal rsuls ar hs horcal rsarch paprs: [5] [] [8] [8] ad [3] L f ( m b a ral-valud arhmcal fuco whos doma s posv gr { } W say ha srogly addv fucos sasfs h followg rsrco ( f ( m + f ( m + f ( whvr ( m k ( f ( p f ( p for all prm p ad k For ampl l f ( m b h umbr of prm dvsors of m Ad h addv fucos jus sasfy ( Th w hav h followg rsuls du o Bkls [3] I Bkls s work h gv h cssary ad suffc codos for h wak covrgc of h dsrbuo fucos of srogly addv fucos o h f Posso law covoluos Thorm 4: L f( m ( b a srogly addv fuco dpd o Dfd h pmf of u by v ( f ( m u v ( f ( m u #{ m : f ( m u} [ ] L fucos f( m o ach prm umbr p s valus from h ogav gr s If { c c c } < c < c < < c r r ( b as dscrbd abov ad h lmd pgf of v ( f ( m u s r c u λ ( s v f( m u s ( s u lm ( f umbrs c ( j r ar larly dpd j ovr h fld Q h codos (9 ar sasfd ( hy ar cssary ad suffc hs cas Th proof of Thorm 4 ad Thorm 5 s [3] h rwr h corrspodg characrsc fuco o h pgf of v ( f ( m u Šaulys [8] prov ha f f s a srogly addv fuco o prm umbr wh f ( p {} h f s Posso dsrbud o h grs udr h codo ha r (9 Ths s a spcal cas of Thorm 4 Numrcal ampl: L f ( m f ( p b h sum of f( p udr codo ha p m whr pm p < l l or p > (l l f( p l l p (l l 4 (l l p (l l Bkls [3] oba λ lm l λ lm l p p p p f ( p f ( p Thus u approma o Hrm dsrbuo wh pmf u u [ ] [ ] u λ λ ( λ + λ u (l v( f( m u ( u 4 ( u Frqu umbr Tabl Frqu wh Thorcal frqu Frqu ( Frqus ( 4 Frqus ( l p lm ma ; lm ; p f ( p p pl p f ( p c p f ( p lm λ ( j r p h h lmd pgf of v ( f ( m u s r c u λ ( s v f( m u s ( s u lm ( Hc u CP( λ λ λr Thorm 5: L srogly addv fucos f ( m ( Noc ha frqus udr a suffc larg ar o qualy horcal frqus Tabl (jus u 3 s accura Th l l 9 s o suffc larg

10 Amrca Joural of Thorcal ad Appld Sascs 3; (5: - 9 I ordr o oba qualy w would hav o l ll som fasho Th covrgc ra s vry slow Sc lmd codos compur w ca calcula h largr valu of hs papr by Malab Thorm 6 (Erdos-Wr 939: A cssary ad suffc codo for a ral addv fuco f ( m o hav a lmg dsrbuo s ha h followg hr srs covrg smulaously for a las o valu of h posv ral umbrr : f ( p f ( p ; ; p p p f ( p > R f ( p R f ( p R Wh hs codos ar sasfd h characrsc fuco of h lm law s gv by h covrg produc v f ( p ( [ p ( p τ v ] p v ϕ τ Th lm law s cssarly pur I s couous f ad oly f p f ( p For furhr plaao for Erdos-Wr horm s [9] ad [9] Usg Erdos-Wr horm w hav h followg horm Proposo I h Erdos-Wr horm f < ad f ( m aks o-gav gr valu p f ( p h ϕ( τ approma o a dscr compoud Posso dsrbuo wh pgf v f ( p v p ( p s p f ( p v f ( p P( s ( s Proof: Rwr h characrsc fuco ϕ( τ o h pgf ad oc ha + for a suffc small v v f ( p ( v p p s h v v f ( p ( [( + ( ] p v P s p p s p v v f ( p [( p ( p s p ] p v + p v f ( p v p ( p s p v v f ( p v p ( p s p f ( p v f ( p 5 (Ss Ivory Polcs udr Surg Posso Dmad Cosdrg h sals of a rprs h rval s a wk ad h wk sals ar radom Th maagm dcds whhr o ordr goods o sasfy h ds of wk O of h mos smpl vory sragy s (ss vory polcs: h lowr boud of s h uppr boud of S Wh h vory a wkd s lss ha s ordr socks o rach S ohrws do ordr I fac w d o ak ordrg f sorag f ou of sock paym ad purchas f o accou ad h formula a vory sragy o mmz h oal avrag cos Grally spakg h arrval of dmad sasfs h propry of dscr saoary dpd crms procss Bu a cusomr may purchas h sam goods mor ha o oc so w ca assum ha h radom dmad sasfs surg Posso dsrbuo Suppos ha ach ordrg f s c h bd of ach good s c h sorag f of ach good s c ad h loss of ach good ha ou of sor s c 3 I ordr o facla compuao w assum ha h wkly sal r s a ogav gr-valud radom varabl ad h pmf s p( r If h sock h d of h wk s ordr quay s u so h sock a h bgg of h wk s + u sorag capacy vry wk s + u r Accordg o (S s vory sragy f s ordr quay u ; f < s u > ad h cosra s + u S Now w drm (S s hrough fgurg ou h mmum of h avrag cos Th sorag cos ad h pcd valu of back-ordr loss a wk s L ( c ( r p ( r dr + c ( r p ( r dr 3 Th avrag cos a wk s h sum of ordr cos bd cos sorag cos ad h back-ordr loss c + cu + L( + u u > J( u L( u Drmao of S: Now w ak h drvav ad scod drvav of h avrag cos fuco abou dj( u + u c + c p( r dr c 3 p( r dr du + u S + u S ( c + c p( r dr ( c c 3 3 dj( u L du h w ca solv ha wh S sasfs S c3 c p( r dr c + c 3 h J( u rachs h mmum Drmao of s: Th drm h ordr umbr s accordg o If maagm dcd o ordr som goods ad h umbr s u sc + u S h oal cos s J c + c ( S + L( S If h maagm dcds o o ordr ay w good

11 Zhag Humg al: O Nogav Igr-Valud Lévy Procsss ad Applcaos Probablsc Numbr Thory ad Ivory Polcs h h oal cos would bj L( I's obvous ha h cos has o m h cosraj J so w hav: c + L( c + cs + L( S F( c + L( [ c + cs + L( S] s s s h L mmum posv roo ha maks F( sasfd Tha s s m{ > F( } Grally spakg w could oly us graphcal mhod (umrcal mhod o fd h mmum posv roo Wh p( r s dscr radom varabl h mhod o drm (Ss s smlar o h coscuv suao h oly chag s o rplac wh Σ For mor formao abou vory sragy of (Ss s Vo [3] ad Porus [6] For sac a day assumg ha h comg cusomrs a sho sor ar Posso dsrbud wh paramr λ 5 Bu bcaus h shos ar durabl goods w ca suppos ha h probably of buyg a par of shos oc for a cusomr s 8 ad h probably of buyg wo par of h sam shos oc s vry small such as 8 If w suppos ha h probably of a cl buyg ( par of shos s 8 w coclud ha h amou of sold shos s surg Posso dsrbud wh paramr α Assum ha ach ordrg f c 4 h purchas prc of ach par of shos c 6 h sorag f of ach par of shos c h ou of sock damag of ach par of shos c 3 Th usg Mapl6 o plo ad solv h quao S p( r dr c c c + c Udr h sam λ s obvously ha wh r subjcs o surg SPD S A 6 s A 43; bu f r subjcs o Posso dsrbuo(pd S B 47 s B I s asly s ha S surg Posso dsrbuo s bggr ha Posso dsrbuo udr h sam λ Th dmads may arrv suprmposo W d o cras sorag o sasfy dmads Hgh-ch ad quafcaoal vory maagm s a ffcv way o rduc rprs cos ad o mprov h qualy of srvc Du o h ucray of cusomr's arrval may cass surg Posso dsrbuo s closr o h ral lf applyg o som mor gral cas Ad ca also b usd ohr sochasc procss wh h characrsc of suprposo such surac clams daa [34] Rfrcs Fgur 4 drm S wh r subjcs o SPD or PD (a (b Fgur 5 Th graph of F( ((a:spd;(b:pd [] Adlso R M Compoud posso dsrbuos Opraoal Rsarch9667(:73-75 [] Bardorff-Nls O E Pollard D G Shphard N Igr-valud Lévy procsss ad low lacy facal coomrcs Quaav Fac ;(4: [3] Bkls D Covoluos of h Posso laws umbr hory I Aalyc & Probablsc Mhods Numbr Thory: Procdgs of h d Iraoal Cofrc Hoour of J Kublus Lhuaa 3-7 Spmbr 996 (Vol 4 p 83 Nhrlads:VSP BV997 [4] Bll E T Epoal polyomals Th Aals of Mahmacs 93435(: 58-77

12 Amrca Joural of Thorcal ad Appld Sascs 3; (5: - [5] Bro J Lévy procsss Lodo: Cambrdg uvrsy prss998 [6] D F B Sulla possbla d valor cczoal pr ua lgg ad crm alaor A dlla Ral Accadma Nazoal d Lc (Sr VI 99:35-39 [7] D F B L fuzo cararsch d lgg saaa Rdcodlla R Accadma Nazoal d Lc (Sr VI 93:78-8 [8] D Kock J M Galambos J Th rmda prm dvsors of grs Procdgs of h Amrca Mahmacal Socy 987(:3-6 [9] Erdös P Wr A Addv arhmcal fucos ad sascal dpdc Amrca Joural of Mahmacs 9396(3:73-7 [] Faa d Bruo CF Sullo svluppo dll fuzo Aal d Scz Mamach Fsch 8556: [] Gallaghr PX O h dsrbuo of prms shor rvals Mahmaka 9763(: 4-9 [] Gallhr H P Mors P M Smod M Dyamcs of wo classs of couous-rvw vory sysms Opraos Rsarch 9597(3: [3] Gravll A Prm dvsors ar Posso dsrbud Iraoal Joural of Numbr Thory 73(:-8 [4] Hagh FA Hadbook of h Posso dsrbuo Los Agls: Wly;967 [5] Halász G O h dsrbuo of addv ad h ma-valus of mulplcav arhmc fucos Suda Sc Mah Hugarca 97;6: 33 [6] Hofma M Übr zusammgsz Posso-Prozss ud hr Awdug dr Ufallvrschrug Dss Mah ETH Zürch Nr 5 Rf: Sar W; Korrf: Nolf P 955 [7] Jáossy L Réy A Aczél J O composd Posso dsrbuos I Aca Mahmaca Hugarca 95(:9-4 [8] Johso N L Kmp A W Koz S Uvara dscr dsrbuos Nw Jrsy: Wly ; 5 [9] Kmp C D Kmp A W Som proprs of h Hrm dsrbuo Bomrka 9655(3-4: [] Khch AY A w drvao of a formula by P Lévy Bull of h Moscow Sa Uvrsy 937:-5 [Russa] [] Khch AY Mahmacal mhods of quug hory Procdgs of h Sklov Isu VA Sklov 95549(:3- [Russa] [] Kolmogorov A N Sulla forma gral d u procsso socasco omogo A Accad Naz Lc 935: [3] Lévy P Sur ls égrals do ls éléms so ds varabls aléaors dépdas Aal dlla Scuola Normal Supror d Psa-Class d Scz 9343(3-4: [4] Lüdrs R D sask dr sl rgss Bomrka 9346(/:8-8 [5] Posso SD Rchrchs sur la probablé ds jugms maèr crmll maèr cvl précédés ds règls géérals du calcul ds probablés Pars: Bachlr837 [6] Porus EL Foudaos of sochasc vory hory Saford: Saford Uvrsy Prss [7] Prvaul N Sochasc Aalyss Dscr ad Couous Sgs Wh Normal Margals Brl: Sprgr 9 [8] Šaulys J Th vo Mss horm umbr hory Nw Trds Probably ad Sascs 99:93-3 [9] Tbaum G Iroduco o aalyc ad probablsc umbr hory Lodo: Cambrdg uvrsy prss995 [3] Vo A F Wagr H M Compug opmal (s S vory polcs Maagm Scc 965(5:55-55 [3] Voov V G Nkul M O powr srs Bll polyomals Hardy-Ramauja-Radmachr problm ad s sascal applcaos Kybrka 9943(3: [3] Wmmr G Alma G Th mulpl posso dsrbuo s characrscs ad a vary of forms[j] Bomrcal joural (8: 995- [33] Whakr J M Th sho ffc for showrs Mahmacal Procdgs of h Cambrdg Phlosophcal Socy Cambrdg Uvrsy Prss 93733(4: [34] Zhag H Chu L Dao Y Som Proprs of h Gralzd Surg Posso Dsrbuo ad Is Applcaos Suds Mahmacal Sccs 5( :-6