ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS

Transcription

1 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS MARKUS KUBA ABSTRACT. I this wor we discuss two ur odes with geera weight sequeces A, B associated to the, A α N ad B β N, geeraizig two we ow Póya-Eggeberger ur odes, aey the so-caed sapig without repaceet ur ode ad the OK Corra ur ode. We derive sipe expicit expressios for the distributio of the uber of white bas, whe a bac have bee draw, ad obtai as a byproduct the correspodig resuts for the Póya-Eggeberger ur odes. Moreover, we show that the sapig without repaceet ur odes ad the OK Corra ur odes with geera weights are dua to each other i a certai sese. We discuss extesios to higher diesioa sapig without repaceet ad OK Corra ur odes, respectivey, where we aso obtai expicit resuts for the probabiity ass fuctios, ad aso a aaog of the duaitiy reatio. Fiay, we derive iit aws for a specia choice of the weight sequeces.. INTRODUCTION I this wor we aayse two sequetia ur odes with two types of bas. At the begiig, the ur cotais white ad bac bas, with,. Associated to the ur ode are two sequeces of positive rea ubers A α N ad B β N. At every step, we draw a ba fro the ur accordig to the uber of white ad bac bas preset i the ur with respect to the sequeces A, B, subect to the two odes defied beow. The choose ba is discarded ad the sapig procedure cotiues uti oe type of bas is copetey draw. Ur ode I Sapig with repaceet with geera weights. Assue that white ad bac bas are cotaied i the ur, with arbitrary, N. A white ba is draw with probabiity α /α + β, ad a bac ba is draw with probabiity β /α + β. Additioay, we assue for ur ode I that α 0 β 0 0. Ur ode II OK Corra ur ode with geera weights. For arbitrary, N assue that white ad bac bas are cotaied i the ur. A white ba is draw with probabiity β /α + β, ad a bac ba is draw with probabiity α /α + β. The absorbig states, i.e. the poits where the evoutio of the ur odes stop, are give for both ur odes by the positive attice poits o the the coordiate axes {0, } {, 0 }. These two ur odes geeraize two faous Póya-Eggeberger ur odes with two types of bas, aey the cassica sapig without repaceet I, ad the so-caed OK-Corra ur ode II, described i detai beow. We are iterested i the distributio of the rado variabe X, coutig the uber of white bas, whe a bac bas have bee draw. For both ur odes we wi derive expicit expressio for the probabiity ass fuctio P{X, } of the rado variabe X,. Moreover, we wi describe a certai duaity betwee the sapig without repaceet ur ode ad the OK Corra ur ode usig a appropriate trasforatio of weight sequeces... Weighted attice paths ad ur odes. Cobiatoriay, we ca describe the evoutio of the ur odes I ad II by weighted attice paths. If the ur cotais white bas ad bac bas ad we seect a white ba, the this correspods to a step fro, to,, to which the weight α /α + β is associated for ur ode I, ad β /α + β for ur ode II. Coversery, if we seect a bac ba, this correspods to a step fro, to,, to which the weight β /α + β is associated for ur ode I, ad α /α + β for ur ode II. The weight of a path after t successive draws cosists of the product of the weight of every step. By this correspodece, the probabiity of startig at, ad edig at, is equa to the su of the weights of a possibe paths startig at state, ad edig at state,, for, N 0. The expressios for the probabiity, obtaied by this cobiatoria ode are, athough exact, very ivoved ad ot usefu for i.e. the cacuatio of the oets of X,, or for derivig asyptotic expasios for arge ad. Date: May 9, Matheatics Subect Cassificatio. 05A5,60F05. Key words ad phrases. Ur odes, OK Corra, Sapig without repaceet, sequetia ur ode, Jacobi theta fuctio. This wor was supported by the Austria Sciece Foudatio FWF, S9608-N3.

2 2 M. KUBA.2. Coectio to ordiary Póya-Eggeberger ur odes. The dyaics of ordiary Póya-Eggeberger ur odes are very siiar to the dyaics of the two previousy descirbed ur odes. At the begiig, the ur cotais white ad bac bas. At every step, we choose a ba at rado fro the ur - a white ba is draw with probabiity / + ad a bac ba is draw with probabiity / + - ad exaie its coor ad put it bac ito the ur ad the add/reove bas accordig to its coor by the foowig rues. If the ba is white, the we put a white ad b bac bas ito the ur, whie if the ba is bac, the c white bas ad d bac bas are put ito the ur. The vaues a, b, c, d Z are fixed iteger vaues ad the ur ode is specified by the trasitio atrix M a b c d. We ca speciaise the sequeces A α N ad B β N i order to obtai two faiies of Póya-Eggeberger ur odes as specia cases..3. Sapig without repaceet ad geeraizatios. This cassica exape correspods to the ur with trasitio atrix M 0 0. I this ode, bas are draw oe after aother fro a ur cotaiig bas of two differet coors ad ot repaced. What is the probabiity that bas of oe coor reai whe bas of the other coor are a reoved? The geeraized sapig without repaceet ur is specified by ba trasitio atrix give by M a 0 0 d, with a, d N. I the geera ode oe ass for the uber of white bas whe a bac bas have bee reoved, startig with a white bas ad d bac bas. Let Y a,d deote the rado variabe coutig the uber of white bas whe a bac bas have bee reoved. The rado variabe Y a,d is reated to X, of ur ode I i the foowig way. Y a,d a X,, for the specia choices A α N a N, B β N d N. The ost basic case a d ca be treated by severa differet ethods such as geeratig fuctios, attice path coutig arguets, etc. The distributio of Y, is a foore resut ad give as foows. P{Y, } + +, The OK Corra probe ad geeraizatios. The so-caed OK Corra ur serves as a atheatica ode of the historica gu fight at the OK Corra. This probe was itroduced by Wiias ad McIroy i [22] ad studied recety by severa authors usig differet approaches, eadig to very deep ad iterestig resuts; see [20, 0,, 2, 9, 3]. Aso the ur correspodig to the OK corra probe ca be viewed as a basic ode i the atheatica theory of warfare ad coficts; see [, 2]. A iterpretatio is give as foows. Two groups of gue, group A ad group B with ad gue, respectivey, face each other. At every discrete tie step, oe gua is chose uifory at rado who the shoots ad is exacty oe gua of the other group. The gufight eds whe oe group gets copetey eiiated. Severa questios are of iterest: what is the probabiity that group A group B survives, ad what is the probabiity that the gufight eds with survivors of group A group B? Apparety, the described process correspods to a Póya-Eggeberger ur ode with trasitio atrix M 0 0. This ode was aayzed by Wiias ad McIroy [22], who obtaied a iterestig resut for the expected vaue of the uber of survivors. Usig artigae arguets ad the ethod of oets Kiga [0] gave iitig distributio resuts for the OK Corra ur ode for the tota uber of survivors. Moreover, Kiga [] obtaied further resuts i a very geera settig of Lachester s theory of warfare. Kiga ad Voov [2] gave a ore detaied aaysis of the baaced OK Corra ur ode usig a igeious coectio to the faous Frieda ur ode; aogst others, they derived a expicit resut for the uber of survivors. I his Ph. D. thesis [9] Puyhaubert, guided by Faoet, exteded the resuts of [0, 2] o the baaced OK Corra ur ode usig aaytic cobiatoric ethods cocerig the uber of survivors of a certai group. His study is based o the coectio to the Frieda ur showed i [2]. He obtaied expicit expressio for the probabiity distributio, the oets, ad aso reobtaied ad refied ost of the iitig distributio resuts reported earier. Soe resuts of [9] where reported i the wor of Faoet et a. [3], see aso Faoet [2]. Stade [20] obtaied severa iitig distributio resuts for the geeraized OK Corra ur, as itroduced beow, ad aso for reated ur odes with ore geera trasitio probabiities. A atura geeraizatio of the OK Corra ur ode is to cosider a ur ode with trasitio atrix give by M 0 b c 0, b, c N, which ay serve as a sipe atheatica ode of warfare with uequa fire power. Let Y c,b deote the rado variabe coutig the uber of white bas whe a bac bas have bee reoved, startig with c white bas ad b bac bas. The rado variabe Y c,b is reated to the rado variabe X,

3 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS 3 of ur ode II i the foowig way. Y c,b c X,, for the specia choices A α N c N, B β N b N. Cocerig the specia case c b Puyhaubert ad Faoet obtaied the foowig resut for the distributio of Y,.! + r P{Y, } r r +,. +! r r.5. Notatio. Throughout this wor we use the otatios N {, 2, 3,... } ad N 0 {0,, 2,... }. Furtherore, we deote with i the iagiary uit, i 2. We use the abbreviatios x s : xx x s + for the faig factorias. Further we deote with [ { ] the usiged Stirig ubers of the first id ad with } the Stirig d ubers of the secod id. We use the otatio X X for the wea covergece, i. e. the covergece i distributio, of the sequece of rado variabes X to a rado variabe X..6. Overview. We derive expicit foruas for the probabiity ass fuctios of the two ur odes I ad II for two give sequeces A, B of positive ubers A α N ad B β N assuig that both sequeces A, B satisfy the coditios α α ad β β, < <. Our expicit resuts cocerig the distributio of X, ad the resutig expressios for the Póya-Eggeberger ur odes geeraize the correspodig resuts of [3, 8]. Furtherore, we aso study higher diesioa ur odes with r 2 types of bas. Our approach cosists of costructig a faiy of fuctios f, t, N, which ca be reated to the probabiity geeratig fuctio of the rado variabes X,. It is based o the wor of Stade [20], see aso Kiga ad Voov [2]. Stade did derive a itegra represetatio for the probabiites P{X, } for ur ode II. I this wor we revisit Stade s approach ad exted it to ur ode I i order to obtai itegra represetatios for both ur odes, which i tur wi be used to derive the expicit expressios for the probabiity ass fuctio P{X, }. Furtherore, cocerig the Póya-Eggeberger versios of ur odes I ad II we wi derive various expressios for the oets of the rado variabes X, or ore precisey Y a,d ad Y c,b. Later, we exted ur odes I ad II to higher diesios by cosiderig r 2 differet types of bas. Moreover, we wi prove that both ur odes are dua to each other i a certai sese, see Sectio 6, which aso exteds to the higher diesioa ur odes. Fiay, i Sectio 7 we study ur ode I with weight sequeces α N N ad β N 2 N ad show that iterestig iitig distributios arise, curiousy ivovig the Jacobi Theta fuctio Θq Z q Resuts for ur ode I. 2. MAIN RESULTS Theore. The probabiity ass fuctio of the rado variabe X,, coutig the uber of reaiig white bas whe a bac bas have bee draw i ur ode I with weight sequeces A α N, B β N Sapig with repaceet with geera weights, is for, ad 0 give by the expicit forua P{X, } β h h β h h h+ h+ α h α h α i + β β i β assuig that α α ad β β, < <, ad that α 0 0. α α i, i β i + α Adaptig the weights sequeces A, B to the specia case A a N ad B d N, with a, b N, correspodig to the geeraized sapig Póya-Eggeberger ur ode with ba trasitio atrix M a 0 0 d, eads to the foowig resut.

4 4 M. KUBA Coroary. The probabiity ass fuctio of the rado variabe Y a,d, coutig the uber of reaiig white bas whe a bac bas have bee reoved i the geeraized sapig without repaceet Póya-Eggeberger ur ode with ba trasitio atrix M a 0 0 d, startig with a white ad d bac bas, is give by P{Y a,d a} + d a + d a + a, 0. The oets of the oraized rado variabe Ŷa,d Y a,d /a are give by the foowig expressios. s E Ŷ a,d s + as, E s { } s Ŷa,d s d + a, d 0 where { } deote the Stirig uber of the secod id, ad x s xx x 2... x s the faig factorias Resuts for ur ode II. Theore 2. The probabiity ass fuctio of the rado variabe X,, coutig the uber of reaiig white bas whe a bac bas have bee draw i ur ode II with weight sequeces A α N, B β N OK Corra ur ode with geera weights, is for, ad give by the expicit forua P{X, } α α α α α + β + d h α + β h β h, + α β β h assuig that α α ad β β, < <. For, ad 0 we obtai P{X, 0} α α h α + β + h α + β h β h. + α β β h Adaptig the weights sequeces A, B to the specia case A c N, B b N, with b, c N correspodig to the geeraized OK-Corra Póya-Eggeberger ur ode with ba trasitio atrix M 0 b c 0, we obtai the foowig resuts. Coroary 2. The probabiity ass fuctio of the rado variabe Y c,b, coutig the uber of reaiig white bas whe a bac bas have bee reoved i the geeraized OK-Corra Póya-Eggeberger ur ode with ba trasitio atrix M 0 b c 0 startig with c white ad b bac bas, is give by b P{Y c,b c} +!! c + b +, c + c!! b + c + for, b 0 b P{Y c,b 0}!! c + b +, for 0. Usig the expicit expressios for the probabiities ad a iportat resut of Puyhaubert [9] ad Faoet [2], origiay deveoped for the aayis of the case b c, we ca give a precise descriptio of the oets. c h

5 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS 5 Coroary 3. The oets of the rado variabe Ŷc,b Y c,b /c, where Y c,b couts the uber of reaiig white bas whe a bac bas have bee reoved i the geeraized OK-Corra Póya-Eggeberger ur ode with ba trasitio atrix M 0 b c 0 startig with c white ad b bac bas, are give by the exact forua EŶ c,b s c + + +! b + c + f s+ Q + g s+, 0 where f u![z ]F z, u, g u![z ]Gz, u, with F z, u e u e z +z, Gz, u e u ad Q degree 2s such that i0 i i EM s Ŷc,b s!2s +! b e z +z z 0 ue t e u e t +t dt, deotig Raaua s Q-fuctio. Moreover, there exists oic poyoias M s s N of c b c b ++s s!2s!! c b 0 + c ++s, b where the coefficiets i,s, with i 2s, of M s are specified by the triaguar syste of iear equatios 2s i i,s f i+ X 0, 2s i i,s g i+ X s!2 s X s+. 3. DERIVING THE EXPLICIT RESULTS: URN MODEL I By defiitio of ur ode I the probabiities P{X, } satisfy the foowig syste of recurrece reatios. α P{X, } P{X, } + β P{X, },,, 0 α + β α + β P{X,0 } δ,,, 0, ad P{X 0, } δ,0,, 0. I order to prove the resut above we first derive a cass of fora soutio of the syste of recurrece reatios 2. The, out of the cass of fora soutios we deterie the proper soutio by adaptig to the iitia coditios P{X,0 } for 0. We wi preset the proof usig a sequece of eata. Lea. The quatities f,, t C, defied as ξ t t α f,, t t α + t with paraeter ξ t C, satisfy for each fixed t i R ad, ad a recurrece reatio of the type 2. Moreover, f,, t is a eroorphic fuctio i C, with poes at the vaues { β } ad {α }. β, 2 Proof. We observe that for arbitrary,, f,, t f,, t t α, f,, t f,, t + t. β Cosequety, α f,, t + β f,, t f,,t α t + β + t f,, t, α + β α + β α + β which proves that f,, t satisfies a recurrece reatio of the type 2. Note that we excude the case 0, sice we assue that α 0 0. Sice a of the quatities f,, t C are soutios of a recurrece reatio of the type 2, we ca derive aother faiy of soutios.

6 6 M. KUBA Lea 2. The vaues F,, i i f,,tdt are for,, N fora soutios of the syste of recurrece reatios 2, with paraeter ξ C assued to be a costat idepedet of t. For ad we ca obtai F,, by two differet cotours of itegratio, F,, f,, tdt f,, tdt, with C γ γ 2, C 2 γ γ 3, C C 2 where γ z iz, R z R, γ 2 z Re iϕ, π/2 ϕ 3π/2, γ 3 z Re iπ ϕ, π/2 ϕ 3π/2, such that R > ax, {β, α }. Proof. By defiitio of f,, t the first stated asseratio easiy foows. By Lea f,, t is a eroorphic fuctio i C, with sipe poes at the vaues { β } ad {α }. By taig the iit R i the two differet cotours of itegratio we observe that the cotributios of γ 2 ad γ 3 are zero i the iit, ad oy the cotributio of γ reais. Hece, i i f,, tdt C f,, tdt C 2 f,, tdt. Next we wi derive a aterative represetatios for F,, usig the fora residue cacuus. Lea 3. Assue that the weight sequece A α N satisfies α i α for a i, N with i. The the vaue F,, adits for ad, the foowig represetatio. F,, ξ β h α h h h α α. i β i + α Assue that the weight sequece B β N satisfies β i β for a i, N with i. The the vaue F,, adits for ad, the foowig represetatio. F,, ξ β h α h h h α i. + β β i β Proof. By defiitio of F,, ad Lea 2 we have F,, ξ f,, tdt ξ C 2 β h h h ξ α h C 2 C 2 dt t α α t dt i + t β, β + t By the residue theore with respect to the poes α,, we obtai the represetatio F,, Resf,,, α h ξ β h α h, h h h α α i β i + α by the assuptio o the weight sequece A. A siiar arguet eads to the secod represetatio for F,,. Note that oe ca aterativey covert betwee the two expressio usig the partia fractio idetity α + x h x + α h α α h. 3 h For exape, appyig the partia fractio idetity to the secod expressio gives F,, ξ β h α h i. α α h β + α h β i β h h Aother appicatio of the partia fractio idetity provides the first expressio. h i

7 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS 7 We sti have to adapt the paraeter ξ i the represetatio of F,, to the iitia coditios P{X,0 } δ, for, 0, ad the chec the iitia coditios. First we wi cosider the specia vaue F,, i order to choose the paraeter ξ i the represetatio of F,, based o the expicit expressio. The we wi chec if the iitia coditios are fufied. Let. We have F,, ξ h β h α Appicatio of 3 with respect to the case gives i. α + β β i β ξ h β h α F,, h β h + α. Settig 0 we obtai F,0, ξ α. I order to adapt to the specia cases of the iitia coditios P{X,0 } for we choose ξ /α for ad. The we exted the defiitio of f,, t ad F,, to cope with the additioa boudary cases 0, 0 ad 0. We obtai the desired represetatio by a sight odificatio of the defiitio of f,, t i the case of 0. Lea 4. For, ad 0 the vaues P{X, } ˆf,, C 3 tdt, with ˆf,, t α t α t α t t α + t β + t β i, for,, for 0, where C 3 γ γ 2 γ 3 γ 4, γ z iz, R z ε, γ 2z εe i3π/2 ϕ, 0 ϕ π, γ 3z iz, ε z R ad γ 4z Re iπ ϕ, π/2 ϕ 3π/2, such that R > ax, {β, α } ad ε < i, {β, α }, are the proper soutio of the syste of recurrece reatios 2 fufiig the stated iitia coditios ivodig the boudary cases 0 ad 0. Proof. First we chec that the iitia coditios P{X,0 } δ, for ad 0. For we ca use our previous coputatio which ead to the choice ξ /α, ad the stated resut easiy foows usig 3. For we use residue theore ad the the partia fractio decopositio x x + α x + α +... x + α α x + α i i. α i α Taig the iit x 0 i the idetity above provides the eeded resut. Fiay, we chec the iitia coditios P{X 0, } δ,0 for N ad N 0. For 0 we iediatey obtai P{X 0, 0}, due to the additioa sipe poe at t 0. Cocerig the reaiig cases we rey o the itegra represetatio of the probabiities ˆf,, C 3 tdt. By Lea ˆf,, t has o siguarities with egative rea part ore precisey, it has reoveabe siguarities at {α }. Hece, by the Cauchy itegra theore P{X 0, } C 3 ˆf0,, tdt 0 for ad 0. Rear. It sees difficut prove that the iitia coditios P{X 0, } δ,0 for N ad N 0 are satisfied reyig o the other itegra represetatio, or soey o the two expicit expressios vaid for ad. Proof of Theore. By Lea 3 we obtai for, ad the stated expicit expressios. Our subsequet cacuatios show that stated expressios are aso vaid for 0. By Lea ad Lea 4 the recurrece reatios ad the iitia coditios are are fufied. It reais to prove that the stated foruas are aso vaid i the case 0.

8 8 M. KUBA By the residue theore we obtai P{X, 0} C ˆf,,0 tdt h β h h α h 0 α α h β h h α h 0 α α h β h α α α i β i + α, i β i + α h α h i β i + α subect to α 0 0, which proves the first part of the stated resut for 0. The secod forua foows easiy usig the partia fractio decopositio 3. Aterativey, oe ay chage the cotour of itegratio C 3 ad coect the residues at the poes β,..., β. Proof of Coroary. I order to obtai the resuts for the Póya-Eggeberger geeraized sapig ur ode with ba trasitio atrix M a 0 0 d, we sipe adapt the weights sequeces A, B to the specia case A a N ad B d N, with a, b N. Cocerig the oets we use the we ow, see e.g. [7], basic idetities x + x + +, x + x x Let Ŷa,d deote the scaed rado variabe Y a,d /a. The factoria oets of Ŷa,d are derived as foows. EŶ s a,d + d s P{X a,d a} a s. 0 0 s + d a By iterchagig suatio ad usig the first idetity stated i 4 we get further EŶ s a,d +s+ d a s s + d a s s s + d. a + d a The appicatio of the secod idetity stated i 4 ad the fact that s { } s EŶ a,d s EŶ a,d, 5 where { } deotes the Stirig ubers of the secod id ead obtai the foowig resuts. s { } EŶ s a,d s s + as, EŶ a,d s d + a. d 0 4. DERIVING THE EXPLICIT RESULTS: URN MODEL II The derivatio of the expicit resuts for ur ode II is siiar to are previous approach to ur ode I; therefore we wi be ore brief. By defiitio the probabiities P{X, } satisfy the foowig syste of recurrece reatios. β α P{X, } P{X, } + P{X, },,, 0 α + β α + β P{X,0 } δ,, ad P{X 0, } δ,0,. As before, we first derive a cass of fora soutio of the syste of recurrece reatios 6. The, out of the cass of fora soutios we deterie the proper soutio by adaptig to the iitia coditios P{X,0 } for 0. 6

9 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS 9 Lea 5. The quatities g,, t C, defied as ξ α t g,, t α, t + β t with paraeter ξ C, satisfy for each fixed t i R ad, ad a recurrece reatio of the type 6. Moreover, g,, t is a eroorphic fuctio i C, with poes at the vaues { /β } ad {/α }. Proof. We observe that for arbitrary,, g,, t g,, t α t, g,, t g,, t + β t. Cosequety, β α g,, t + g,, t g,,t β β α t + α + α β t g,, t, α + β α + β α + β which proves that g,, t satisfies a recurrece reatio of the type 2. Next we derive aother faiy of soutios. Lea 6. The vaue G,, i i g,,tdt is for,, N a fora soutio of the syste of recurrece reatios 6, with paraeter ξ C assued to be a costat idepedet of t. For ad we ca obtai G,, by two differet cotours of itegratio, G,, g,, tdt g,, tdt, with C γ γ 2, C 2 γ γ 3, C C 2 where γ z iz, R z R, γ 2 z Re iϕ, π/2 ϕ 3π/2, γ 3 z Re iπ ϕ, π/2 ϕ 3π/2, such that R > ax, {/β, /α }. Proof. By defiitio of g,, t the first stated asseratio easiy foows. By Lea 2 g,, t is a eroorphic fuctio i C, with sipe poes at the vaues { /β } ad {/α }. By taig the iit R i the two differet cotours of itegratio we observe that the cotributios of γ 2 ad γ 3 are zero i the iit, ad oy the cotributio of γ reais. Hece, i i g,, tdt C g,, tdt C 2 g,, tdt. Next we wi derive a aterative represetatios for G,, usig the fora residue cacuus. Lea 7. Assue that the weight sequece A α N satisfies α i α for a i, N with i. The the vaue G,, adits for ad, the foowig represetatio. G,, ξ α α α α +. h α + β h Assue that the weight sequece B β N satisfies β i β for a i, N with i. The the vaue G,, adits for ad, the foowig represetatio. β + G,, β h. + α β β h Proof. By defiitio of G,, ad Lea 2 we obtai the stated resuts by the residue theore. We adapt the paraeter ξ i the represetatio of G,, to the iitia coditios P{X,0 } δ, for ad 0 ad obtai after a few cacuatios the ecessary coditio ξ α for. h

10 0 M. KUBA Lea 8. For, ad 0 the vaues P{X, } α αt ĝ,, t αt +β t t αt +β t C 3 ĝ,, tdt, with, for,, for 0, where C 3 γ γ 2 γ 3 γ 4, γ z iz, R z ε, γ 2z εe i3π/2 ϕ, 0 ϕ π, γ 3z iz, ε z R ad γ 4z Re iπ ϕ, π/2 ϕ 3π/2, such that R > ax, {/β, /α } ad ε < i, {/β, /α }, are the proper soutio of the syste of recurrece reatios 2 fufiig the stated iitia coditios ivodig the boudary cases 0 ad 0. Proof. The iitia coditios P{X,0 } δ, for ad are satisfied sice ξ α. Fiay, we chec the iitia coditios P{X 0, } δ,0 for N ad N 0. For 0 we otai P{X 0, 0} due to the additioa sipepoe at t 0. Cocerig the reaiig cases we rey o the itegra represetatio of the probabiities C 3 ĝ,, tdt. By Lea ĝ,, t has o siguarities with egative rea part ore precisey, it has reoveabe siguarities at {/α }. Hece, by the Cauchy itegra theore P{X 0, } C 3 ĝ 0,, tdt 0 for ad 0. Fiay, the probabiities P{X, 0}, with, 0 are cacuated i two differet ways usig the residue theore. Proof of Coroary 2. I order to obtai the resuts for the Póya-Eggeberger geeraized sapig ur ode with ba trasitio atrix M 0 b c 0, we adapt the weights sequeces A, B to the specia case A c N ad B b N, with a, b N. Cocerig the oets of Ŷc,b Y c,b /c we proceed as foows. We obtai the ordiary oets EŶ c,b s with s, usig the expicit expressios for the probabiities P{Y c,b c} by suatio. E Ŷc,b s s s+ c P{Y c,b c}!! b + c +. Iterchagig the order of suatio eads to E Ŷc,b s c b 0 +! c b 0 + c b +! c b 0 s+! + The secod su was studied i detai i [9]. We wi use the foowig iportat resut. 0 b! s. Lea 9 Puyhaubert [9]. Let the fuctios F z, u ad W z, u be defied as z F z, u e u e z +z, Gz, u e u e z +z ue t e u e t +t dt. Furtherore, et f u![z ]F z, u ad g u![z ]Gz, u, where the poyoia f has degree 2 ad g has degree + 2. The eadig coefficiet of f 2 is give by 2!! ad the eadig coefficiet of g 2+ is give by 2!! Moreover, et Q deote Raaua s Q-fuctio defied as Q ! i i. i0 The su! s ca be represeted i ters of f, g ad Q as foows.! s f s+q + g s+.

11 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS The resut above is prove i Puyhaubert [9], p. 7 8, we refer the reader to the proof therei. As poited out i [9] the fuctio F z, u is cosey reated to Stirig ubers of the secod id. Note that i [9] there is a tiy isprit, aey the factor! i the defiitio of f ad g is issig. For a coprehesive discussio of Raaua s Q-fuctio we refer the reader to the wor of Faoet et a.[5]. Usig Lea 9 we iediatey obtai the stated descriptio of the oets of of Ŷc,b. Furtherore, the secod assertio foows easiy by copariso of the degrees i f X ad g X, siiar to [9]. The secod expressio for EM s Y c,b is easiy obtaied by sipe aipuatios of the bioia coefficiets. 5. EXTENSION TO HIGHER DIMENSIONAL MODELS It is atura to as whether the sipe expicit expressios for the probabiities P{X, }, as give i Theore, ca be exteded to higher diesioa odes, i.e. ivovig ore tha two bas. I the foowig we wi geeraize ur odes I ad II by cosiderig urs with r 2 differet types of bas. At the begiig, the ur cotais bas of type bas, with for r. Subsequety, we wi use the utiidex otatio,..., r. Associated to the ur ode are r sequeces of positive ubers A [] α [] N, for r. At every step, we draw a ba fro the ur accordig to the ubers of bas preset i the ur with respect to the sequeces A [] r, subect to the two odes defied beow. The choose ba is discarded ad the sapig procedure cotiues uti type r of bas is copetey draw. We are iterested i the oit distributio of type up to type r bas whe a type r bas have bee draw.this correspods agai to the absorbig regio A A A 2, with A {,..., r, 0 :,..., r 0} ad A 2 {0,..., 0, r : r } We study the rado vector X X [],..., X [r ], where X [] X [],..., r couts the uber of type bas, startig with bas of types,..., r, i. e. bas of uits, r. Subsequety, we wi derive expicit expressios for the probabiities P{X }, for,..., r, with 0, for the foowig two ur odes, extedig our previous resuts correspodig to the case r 2. Ur ode I. Assue that the ur cotais bas of type bas, r, with r > 0 ad at east oe of the greater tha zero, r. A ba of type is draw with probabiity α [] / r α[], r, with the additioa assuptio that α [] 0 0 for r. Ur ode II. Assue that the ur cotais bas of type bas, r, with r > 0 ad at east oe of the greater tha zero, r. A ba of type is draw with probabiity δ,0 r α [] δ,0 / r h δ h,0 r α [] δ,0. h Note that ur ode I is the atura extesio of the ur ode I with two types bas. By suitabe choice of the sequeces A [] α [] N a N, for r, we obtai the stadard Póya-Eggeberger ur with r differet id of bas, ad ba trasitio atrix M give by M a a a r. 7 I cotrast the extesio of ur ode II is o-stadard i the sese that for ay specificatio of the sequeces A [] α [] N, r, we do ot obtai ordiary higher diesioa Póya-Eggeberger ur odes ayore. The reaso for this wi becoe cear i Sectio 6, where we ucover the duaity reatio betwee these two ur odes. 5.. Expicit resuts for the probabiities. Theore 3. The distributio of the rado vector X, coutig the uber of reaiig type,..., r bas whe a type r bas have bee draw i ur ode I, is for ad 0, with r, give by the expicit

12 2 M. KUBA forua assuig that α [] h P{X } r r r α[], h < <, r. r r f α[r] f r f α[r] f + r α[] h + α[] h r h α [] h α [] h Coroary 4. The distributio of the rado vector Y a Y a [],..., Y a [r ], coutig the uber of type bas, r, i the sapig without repaceet Póya-Eggeberger ur ode with r differet types of bas 7, whe a type r bas have bee draw, startig with a bas of type, r, is give by r r P{Y a a} r+ r a f. r f r ar r r Y The ixed factoria oets E a [] s a of Y a are give by the exact forua E r Ya [] s a r s r+ r a f s f ar r. Theore 4. The distributio of the rado vector X, coutig the uber of reaiig type,..., r bas whe a type r bas have bee draw i ur ode II, is for ad, with r, give by the expicit forua P{X } assuig that α [] h r r r r r f α[] α[], h < <, r. r r α[] α[] +r + α [r] r r f g α[] r α [g] g, h α [] α [] h h Siiar but sighty ore ivoved expressios exist for ur ode II i the cases 0, for r Derivig the expicit resuts. Subsequety, we wi briefy discuss the proof of Theore 3 ad coet o the proofs of Coroary 4 of Theore 4, which are extesios of the proofs of the two diesioa cases, r 2. By defiitio of ur ode I the probabiities P{X } satisfy the foowig syste of recurrece reatios for r, h 0, h r, with at east oe of the h > 0. P{X } r α [] r α[] P{X e }. 8 where e deotes the -th uit vector, with respect to the additioa assuptio that α [] 0 coditios are give by r P{X,..., r,0 } δ,, 0, 0, with at east oe > 0, r P{X 0,...,0,r } δ,0, r, 0. 0 for r. The iitia I order to prove the resut above we first derive a cass of fora soutio of the syste of recurrece reatios. The, out of the cass of fora soutios we deterie the proper soutio by adaptig to the iitia coditios. The crucia part of the proof is the foowig resut. 9

13 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS 3 Lea 0. The quatities f, t C, defied as f, t r ξ t r t α [] t r α [] f t f α [r] h h + r with paraeter ξ t, satisfy for each fixed t i R ad ad, for r, ad r, a recurrece reatio of the type 8. I order to vaidate Lea 0 we observe that r α [] r α [] r f e,t r f, t α[] α[] r f, t α[] r t α [] + α [r] r r α[] α [] t + α α [] [r] r +, f, t + r f t f As i the case r 2 we derive aother faiy of soutios usig the Cauchy itegra theore. α [r] r r f t f α [r] r f, t. Lea. The vaues F, C r... C r f, tdt r... dt are for,, N fora soutios of the syste of recurrece reatios 8, with paraeter ξ assued to be a costat idepedet of t, where the iterior of the cotours of itegratio C are cotaiig the the vaues α [] h, r ad h, pus the origi, ad the orietatio of these cosed curves is atheaticay egative cocwise. It sti reais to adapt to the iitia coditios 9. It turs out that the choice ξ t r α δ,0 t δ,,0 provides the right soutio. Note that ξ t depeds to t, but oy i the case of 0, r. We sip the og ad ivoved cacuatios, which are i pricipe siiar to cacuatios for r 2. Cocerig the resut for the Póya-Eggeberger ur ode with ba trasitio atrix M give by 7 we adapt the sequeces A [] α [] N a N, for r, ad sipify the resut of Theore 3 to obtai the first r Y stateet of Coroary 4. The ixed factoria oets E a [] s a of Y a are derived as foows. E r Ya [] s a 0 s s r r P{Y a a} s r 0 r s s! r r+ r a f s r s f r ar s r s r s s! r r r+ r a f δ,s s r+ r a f s, r s f r ar f ar r r where δ,s deotes the Kroecer-deta fuctio. The resuts for ur ode II are derived aaogousy to the proof of Theore 3 by first derivig a cass of fora soutio of the syste of recurrece reatios ad the adapti to the iitia coditios. 6. DUALITY OF THE TWO URN MODELS Oe ca otice siiarities betwee the expicit resuts for the probabiities P{X, } for ur odes I ad II i the case r 2 ad aso the siiarities of the proofs. However, the higher diesioa odes see to be very differet. We wi show that ur odes I ad II are cosey coected to each other, as they satisfy a certai duaity reatio, ad are essetiay the sae ur ode, aso for the higher diesioa cases.

14 4 M. KUBA Theore 5. Let P{X,,[A,B,I] } deote the probabiity that white bas reai whe a bac bas have bee draw i ur ode I with weight sequeces A α N, B β N ad P{X,,[ Ã, B,II] } the correspodig probabiity i ur ode II with weight sequeces à α N, B β N. The probabiities P{X,,[A,B,I] } ad P{X,,[ Ã, B,II] } are dua to each other, i.e. they are reated i the foowig way. P{X,,[A,B,I] } P{X,,[ Ã, B,II] }, for α α, β β, for a, N. Rear 2. Accordig to the resut above, there is oy oe ur ode urig behid ur odes I ad II, which ca be foruated either as a ur ode of OK Corra type or of sapig without repaceet type with geera weight sequeces. Proof. We observe that the recurrece reatios, as stated i 2, 6 ca be trasfored i the foowig way. P{X,,[A,B,I] } α P{X,,[A,B,I] } + β P{X,,[A,B,I] } α + β α + β β β + P{X,,[A,B,I] } + α β P{X,,[A,B,I] } + β + α β α P{X,,[ Ã, B,II] } + β + α P{X,,[ Ã, B,II] }. α β + P{X,,[A,B,I] } α α β + α P{X,,[A,B,I] } P{X,,[ Ã, B,II] } β + α Sice the iitia coditios coicide we ca trasfor oe ur ode ito the other, which proves the stated resut. I particuar, the duaity reatio expais why ur ode II oos so differet i higher diesios. Theore 6. Let P{X,[A,I] } deote the probabiity that type bas reai whe type r bas have bee draw i ur ode I with weight sequeces A A,..., A r with A α [] N, ad P{X,[ Ã,II] } the correspodig probabiity i ur ode II with weight sequeces à Ã,..., Ãr with à probabiities P{X,[A,I] } ad P{X,[ Ã,II] } are dua to each other, i.e. they are reated i the foowig way. α [] N. The P{X,[A,I] } P{X,[ Ã,II] }, for α[], for a α [], with r. Proof. We observe that the recurrece reatios, as stated i 2, 6 ca be trasfored i the foowig way. P{X,[A,I] } r r α [] r P{X e,[a,i] } α[] r α [] r rh h P{X,[ Ã,II] }, r r α [h] h P{X e,[a,i] } α [] r h α[h] h r α [] r h α[h] r r rh h h P{X e,[a,i] } α [] P{X α [h] e,[ã,ii] } h for > 0, r. Oe sti has to tae ito accouts the various cases of 0, r which eads to the Kroecer deta ters i descriptio of ur ode two. Sice the iitia coditios coicide we ca trasfor oe ur ode ito the other, which proves the stated resut.

15 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS 5 7. DISCUSSION OF A SPECIAL CASE I this sectio we study i ore detai a specia case of ur ode I with respect to the choices A α N N ad B β N 2 N for the weight sequeces A, B. We are iterested i the various iitig distributios arisig for ax{, }. Iterestigy, the Jacobi Theta fuctio Θq Z q 2 aturay arises. Theore 7. The rado variabe X,, coutig the uber of reaiig white bas whe a bac bas have bee draw i ur ode I with weight sequeces A N, B 2 N, satisfies the foowig iitig aws. Case arbitrary but fixed N ad : X, / coverges i distributio to a cotiuous rado variabe Y with support [0, ]: X, d Y, P{Y x} x 0 f qdq, 0 x, with desity fuctio f q beig give by f q q 2. We have aso have covergece of a positive iteger oets of X,, E X, s EY s, EY s! 2 Γ + i sγ + + i s π s sih π s, where i deotes the iagiary uit i 2. 2 Case arbitrary but fixed N ad : X, coverges i distributio to a discrete rado variabe X, Z d Z, P{Z } π sih π + 2, We have aso have covergece of a positive iteger oets of X,, s s { }[ ] s EX, s EZ, s EZ s π sih π, 3 Case i{, } : X, / coverges i distributio to a rado variabe W with support [0, ], whose distributio fuctio ca be expressed i ters of the Jacobi Theta fuctio Θq Z q 2, X, d W, P{W q} Θq, 0 q. We have aso have covergece of a positive iteger oets of X,, X, s E EW s, EW s π s sih π s. We ca sighty exted the previous theore by givig a ore detaied descriptio of the iit aws by studyig the iitig distributios of the rado variabes Y ad Z. Coroary 5. The iit aws of the rado variabe X,, coutig the uber of reaiig white bas whe a bac bas have bee draw i ur ode I with weight sequeces A N, B 2 N, ca be described by the foowig diagra cocerig the covergece i distributio of the various rado variabes. X, oraizatio i{,} Y Z oraizatio W

16 6 M. KUBA Coroary 6. The rado variabes Y, with N, ad W of Theore 7 satify the foowig decopositios. d ɛ Y exp 2, W d ɛ exp 2, where the ɛ are idepedet ad ideticay distributed rado variabes with expoetia distributio Exp. Proofs of Theore 7 ad Coroary 5. I the foowig we wi outie the ai steps of the proofs of Theore 7 ad Coroary 5. Our startig poits are the expicit expressios for P {X, } as give i Theore, P{X, }! 2!! + 2! i i 2 2!2! i i i2 +. By routie aipuatios we derive the aterative represeatio + 2 P{X, } s. Cocerig case such that N is fixed ad we proceed by derivig a oca iit aw, i.e. we study P{X, } for q with q [0, ]. After aipuatio of the stated products we appy Stirig s forua for the Gaa fuctio x x 2π Γx + e x 2x + 288x 2 + O x 3, 0 to the quotiet of Gaa fuctios Γ + Γ + 2 Γ Γ + 2 q 2 + O. Thus we obtai a oca iit theore, which i tur ipies the covergece i distributio after a few routie aipuatios. I order to prove the oet covergece we study first the factoria oets EX, s EX, X,... X, s +. We obtai a cosed for expressio i the foowig way. EX s, s P{X, } 2 0 with s. We use the suatio idetities + X s + s+ ++X +, X + s s + i i + s i 2 2, s + 2, for s N, where the secod idetity easiy foows fro the partia fractio decopositio 3. We obtai the sipe resut EX, s 2 s s 2 s 2 + s. + Cosequety, we ca write the factoria oets as poyoias i of degree s, EX, s 2 s [ ] s 2 s, + s where the [ ] deote the usiged Stirig ubers of the first id, aso ow as Stirig cyce ubers. I order to obtai the ordiary oets we proceed as foows. s s } s s ] EX s, { s } EX, { s h h 2 h 2 + [ ] This ipies that for the ordiary oets satisfy the expasio EX s, s s + Os, { s }[ h h 2 h 2 +.

17 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS 7 which i tur ipies the covergece of the oraized oets. Note that s! 2 Γ + i sγ + + i s π s sih π s, where i deotes the iagiary uit i 2. Cocerig case 2 such that N is fixed ad we sipy observe that the probabiity ass fuctio P{X, } s P{Z }, due to s s π s sih π s. The oet covergece foows iediatey fro the expicit expressio of the ordiary oets of X,. For the fia case 3 i{, } we chec the covergece of the oets usig the expicit expressio X, s E EW s π s i{,} sih π s. By Carea s criterio the oets defie a uique distributio. I order to idetitfy the distributio fuctio we observe, usig Stirig s forua 0, that the probabiity ass fuctio P{X, } satisfies a oca iit aw of the for P{X, } 2 2 q 2 + o Θ q + o. for q with q 0,, where Θq Z q q 2 deotes the Jacobi Theta fuctio. This ipies after a few routie cacuatios, cosistig of derivig the asyptotics of the distributio fuctio P{X, q} q 0 P{X, }, the covergece i distributio of the rado variabe X, / to a rado variabe W with distributio fuctio Θq ad support [0, ]. It is ot iediatey obvious that i q P{W q}. However, by usig the Jacobi Tripe-product-idetity Θq q Z q 2 q 2 q 2 2, we see that P{W q} Θq q2 q 2 2, such that Θq is ideed a we defied distributio fuctio with support [0, ]. Cocerig Coroary 5 we sipy observe that Z d W, ad aso Y d W, which ca be easiy checed fro the expicit expressios of the oets. Proof of Coroary 6. Let ɛ d expλ. The Lapace trasfor Ee sɛ of ɛ is give by Ee sɛ +. Let ɛ s N λ be i.i.d. Exp distributed rado variabes. Usig the fact that c ɛ d Exp c, we obtai E exp s ɛ c Ee sɛ c + c s. Sice the oets rado variabe Y ad W are give by EY s s, EW s π s sih π s s, we obtai the stated decopositios, d Y exp ɛ 2, W d exp ɛ 2.

18 8 M. KUBA Note that such decopositios are cosey reated to Browia excursios ad Zeta fuctios, see the wor of Biae, Pita ad Yor []. There exist fuy aaogous resuts for cosey reated weight sequeces, aso ivovig fuctios reated to the Jacobi Theta fuctio. We wat to poit out two particuar choices, aey A α N N, B β N + 2 N, ad A N, B 2 2 N For exape, choosig the beforehad etioed weight sequeces A N ad B + 2 N, the rado variabe X, / coverges for i{, } i distributio ad with covergece of a iteger oets to a rado variabe W with support [0, ], such that P{W q} q + 2, with oets EW s cosh π 2 Usig a idetity of Jacobi for the cube of the Euer fuctio φq q, aey φ 3 q q + 2, 2sπ. 8s oe ca observe that P{W q} φ 3 q is ideed a we defied distributio fuctio satisfyig P{W }. For the weight sequeces A N ad B 2 2 N oe ecouters a siiar rado variabe W with support [0, ], P{W q} 4 π q 2 2 2, with oets EW s cosh π s. Rear 3. Sice the product r / r + s coverges for arbitrary rea r >, we expect that the resuts of Theore 7 ad Coroary 5 ca be exteded at east to weight sequeces of the for N, r N, for r >. Moreover, decopositios ito sus of expoetiay distributed rado variabes, siiar to Coroary 6, are expected to appear, which are agai cosey reated to the distributios discussed by Biae, Pita ad Yor []. 8. CONCLUSION AND OUTLOOK We have studied two ur odes with geera weights, geeraizig two we ow Póya-Eggeberger ur odes, aey the sapig without repaceet ur ad the OK Corra ur ode. Assuig that the ur cotais two types of bas, we derived expicit resuts for the distributio of the uber of white bas, whe a bac have bee draw. As a byproduct, we obtaied resuts for the aforehad etioed Póya-Eggeberger ur odes. We aso studied higher diesioa ur odes ad aaged to obtai agai expicit expressios. Furtherore, we have show that the two ur odes studied i this paper are essetiay the sae, sice they obey a duaity reatio, aso vaid for the higher diesioa ur odes. Exapariy, we discussed i the preset wor a particuar exape of ur ode I with weight sequeces A N, B 2 N ad obtaied a detaied descriptio of the iit aws, curiousy ivovig the the Jacobi Theta fuctio Θq Z q 2. Sice both ur odes ca be haded together, the author is currety studyig the various iit aws arisig i the Sapig without repaceet/ok Corra ur odes, tryig to uify the previous resuts i iterature. I particuar, the weight sequeces A a N, B b N, with a, b R \ {0} ead to a variety of differet iitig distributios, soe of which are cosey reated to Browia excursios ad the expoetia costructios of Biae, Pita ad Yor []. ACKNOWLEDGEMENTS The author wary thas Aois Pahozer for itroducig hi to this topic, ad for severa vauabe ad iterestig discussios. Moreover, he thas Svate Jaso for poitig out the decopositios of rado variabes Y ad W. REFERENCES [] P. Biae, J. Pita, ad M. Yor, Probabiity aws reated to the Jacobi theta ad Riea zeta fuctios, ad Browia excursios. Bu. Aer. Math. Soc. N.S. 384, eectroic, 200. [2] P. Faoet, Ta: Aaytic Cobiatorics of the Mabiogio ad OK-Corra Urs, 2008 Iteratioa Coferece o the Aaysis of Agoriths, Maresias, Brazi, Sides oie avaiaibe at aofa08.pdf [3] P. Faoet, P. Duas ad V. Puyhaubert, Soe exacty sovabe odes of ur process theory, Discrete Matheatics ad Theoretica Coputer Sciece, vo. AG, 59 8, 2006, i Proceedigs of Fourth Cooquiu o Matheatics ad Coputer Sciece, P. Chassaig Editor. [4] P. Faoet, J. Gabarró ad H. Peari, Aaytic urs, Aas of Probabiity 33, , [5] P. Faoet, P. Graber, P. Kirschehofer ad H. Prodige, O Raaua s Q-fuctio, Joura of Coputatioa ad Appied Matheatics 58,, 03 6, 995.

19 ON SAMPLING WITHOUT REPLACEMENT AND OK-CORRAL URN MODELS 9 [6] P. Faoet ad T. Huiet, Aaytic Cobiatorics of the Mabiogio Ur. Discrete atheatics ad Theoretica Coputer Sciece DMTCS, Proceedigs of Fifth Cooquiu o Matheatics ad Coputer Sciece: Agoriths, Trees, Cobiatorics ad Probabiities, 2008, U. Röser editor, to appear. [7] R. Graha, D. Kuth ad O. Patashi, Cocrete atheatics, Addiso-Wesey Pubishig Copay, Readig, MA, 994. [8] H. K. Hwag, M. Kuba ad A. Pahozer, Aaysis of soe exacty sovabe diiishig ur odes, i: Proceedigs of the 9th Iteratioa Coferece o Fora Power Series ad Agebraic Cobiatorics, Naai Uiversity, Tiai, [9] N. L. Johso ad S. Kotz, Ur odes ad their appicatio. A approach to oder discrete probabiity theory, Joh Wiey, New Yor, 977. [0] J. F. C. Kiga, Martigaes i the OK Corra, Bueti of the Lodo Matheatica Society 3, , 999. [] J. F. C. Kiga, Stochastic Aspects of Lachester s thoery of warfare, J. App. Prob. 39, , [2] J. F. C. Kiga ad S. E. Voov, Soutio to the OK Corra ode via decoupig of Frieda s ur, Joura of Theoretica Probabiity 6, , [3] D. E. Kuth, The toiet paper probe, Aerica Matheatica Mothy 9, , 984. [4] S. Kotz ad N. Baarisha, Advaces i ur odes durig the past two decades, i: Advaces i cobiatoria ethods ad appicatios to probabiity ad statistics, , Stat. Id. Techo., Birhäuser, Bosto, 997. [5] M. Kuba ad A. Pahozer, Liitig distributios for a cass of diiishig ur odes. Subitted. [6] M. Kuba, A. Pahozer ad H. Prodiger, Lattice paths, sapig without repaceet, ad iitig distributios, Eectroic Joura of Cobiatorics, 6, paper R67, [7] H. Mahoud, Ur odes ad coectios to rado trees: a review. Joura of the Iraia Matheatica Society 2, 53-4, [8] N. Pouyae, Cassificatio of arge Póya-Eggeberger urs with regard to their asyptotics. Discrete Matheatics ad Theoretica Coputer Sciece, AD, , [9] V. Puyhaubert, Modées d ures et phéoèes de seui e cobiatoire aaytique, PhD thesis, Ecoe Poytechique, Paaiseau, Frace, [20] W. Stade, Asyptotic probabiities i a sequetia ur schee reated to the atchbox probe, Advaces i Appied Probabiity 30, , 998. [2] D. Stirzaer, A geeraizatio of the atchbox probe, Math. Scietist 3, 04 4, 988. [22] D. Wiias ad P. McIroy, The OK Corra ad the power of the aw a curious Poisso ere forua for a paraboic equatio, Bueti of the Lodo Matheatica Society 30, 66 70, 998. MARKUS KUBA, INSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE, TECHNISCHE UNIVERSITÄT WIEN, WIEDNER HAUPTSTR. 8-0/04, 040 WIEN HTL WIEN 5 SPENGERGASSE, SPENGERGASSE 20, 050 WIEN, AUSTRIA E-ai address: uba@dg.tuwie.ac.at