RATE OF CONVERGENCE OF POLYA S URN TO THE BETA DISTRIBUTION JOHN DRINANE

RATE OF CONVERGENCE OF POLYA S URN TO THE BETA DISTRIBUTION JOHN DRINANE ADVISOR: YEVGENIY KOVCHEGOV OREGON STATE UNIVERSITY ABSTRACT. This pper will document the reserch of the Oregon Stte University 28 Summer REU progrm. The convergence of the Urn to the Bet distribution will be shown. Different methods will be shown to rrive t specific rtes. Also Mthemtic 6 code will be included for generting the distribution of the urn so the reder cn visulize the convergence. 1. INTRODUCTION This project ws motivted from the wor of Kovchegov concerning the behvior of edge reinforced rndom wls. The prticles move long the line, nd the probbility of the prticles movement converges to the Bet Distribution. Two rtes were derived for this process, one of order 1/n 2 nd one rte of the order of 1/ n. 2. BACKGROUND INFORMATION During the course of this pper the reder will need to be fmilir with couple of theorems, reltionships, nd definitions. This section is where these necessities re stted. Berry-Esseen 2.1. [FW] Let X 1,X 2,... be i.i.d with E[X i ],E[X 2 i ]σ2, nd E[ X i 3 ]ρ <. If F n x is the distribution of X 1 + +X n /σ n nd N x is the stndrd norml distribution then F n x N x 3ρ/σ 3 n Crmér s 2.1. [HF] Let X i be i.i.d. R-vlued rndom vribles stisfying ϕt E[e tx 1] < t R Let S n n i 1 X i. Then, for ll > E[X 1 ], 1 lim n n logp[s n n] I where Iz sup[zt logϕt] t R Dte: August 14, 28. This wor ws done during the Summer 28 REU progrm in Mthemtics t Oregon Stte University. 69

7 Drinne 28 Definition of Poly s Urn 2.1. [RS] Suppose tht n urn initilly contins n red nd m blue blls or the initil mounts B,R of blue nd red. At ech stge bll is rndomly chosen, its color is noted, nd it is then replced long with nother bll of the sme color. DeMoivre-Lplce Theorem 2.1. [RS] If S n denotes the number of successes tht occur when n independent trils, ech resulting in success with probbility p, re performed then, for ny < b, Bet Function 2.1. [RS] P Bet Distribution 2.1. [RS] S n np np1 p b βx,y Gmm-Bet Reltionships 2.1. [RS] Φb Φ p x 1 1 p y 1 d p f p 1 βx,y px 1 1 p y 1 Γx + 1 x Γx Sterling s Formul 2.1. [FW] Γz β,b ΓΓb Γ + b z z z 1 1 + O e z

Rte of Convergence of Poly s Urn to the Bet Distribution 71 3. BETA DISTRIBUTION AND THE URN S DISTRIBUTION The distribution of the urn cn be thought of the probbilities of reching certin proportion of red blls in the urn. You cn clculte the distribution by hnd by creting the tree of different possibilities. For exmple given the urn strts with 1 red nd 1 blue bll our urn fter one drw will either hve 2 red nd 1 blue or 1 red nd 2 blue. So the probbility tht our proportion of red is 1/3 is 1/2 nd the probbility of the proportion being 2/3 is lso 1/2. This is how the distribution is built. The following section of grphs re the cumultive distribution grphs of the bet distribution combined with the cumultive distribution of the empiricl distribution for the urn from left to right n5,1,15. These grphs were generted in Mthemtic 6. 1. 1. 1..8.8.8.6.6.6.4.4.4.2.2.2... FIGURE 1. R 1,B 1 1. 1. 1..8.8.8.6.6.6.4.4.4.2.2.2... FIGURE 2. R 1,B 3 1. 1. 1..8.8.8.6.6.6.4.4.4.2.2.2... FIGURE 3. R 3,B 1 Now we cn see the motivtion for both showing this structure converges to the bet distribution nd lso finding the rte t which this occurs.

72 Drinne 28 4. CONVERGENCE OF URN Here we hve tht the probbilities of different rerrngements of drws from the urn s equl. We see this esily by compring two different orders of drwing red nd q blue. P P R R + 1...R + 1B B + 1...B + q 1 R + B R + B + 1...R + B + n 1 Now we wnt to loo t the probbility of outcomes of the urn in terms of choosing reds nd the totl probbility of this which is found by multiplying by the binomil coefficient. n P reds in n trils P }{{} }{{} n Now we re trying to sy something bout the long term behvior of this urn so we should define something tht gives n ide of the urn t nown time step. R n ρ n then, ρ lim R n + B n n R n + B n Now if we condition on certin long term proportion of red blls in the urn then we cn write the probbility of drwing reds in n trils in new wy. P reds in n trils ρ p R n n p 1 p n The term ρ is rndom vrible nd hs density function f p nd cumultive distribution function Fp. If we wnt to find the probbility of reds in n trils without conditioning on certin end proportion we cn integrte over ll p with the density function. Since p is proportion then p 1. 1 P reds in n trils P reds in n ρ p f pd p n 1 p 1 p n f pd p For the next step of the derivtion we need to exmine this probbility in different wy. Noticing there is fctoril type structure ming up the probbility of choosing red in n trils we should try to simplify our expression using gmm functions. 2 P reds in n trils n P }{{} } {{ } n n R R + 1...R + 1B B + 1 B + n 1 R + B R + B + 1 R + B + n 1 n ΓR + ΓB + n ΓR + B ΓR ΓB ΓR + B + n n βr +,B + n 1 βr,b

Rte of Convergence of Poly s Urn to the Bet Distribution 73 Thus setting eqution 1 nd eqution 2 equl we hve n interesting reltionship for our probbility of red in n trils. Which holds for ll n. 3 p 1 p n f pd p βr +,B + n βr,b p 1 p n f pd p p 1 p n f pd p 1 1 p R+ 1 1 p B+n 1 d p βr,b p 1 p n pr 1 1 p B 1 d p βr,b Since 3 is true for ll n then we cn see tht we hve n expression for f p which is the sme for the bet distribution. 4 f p 1 βr,b pr 1 1 p B 1 Now we cn write F ρ nd F ρn where 1. 1 F ρ Pρ βr,b P Rn F ρn Pρ n P R n + B n p R 1 1 p B 1 d p R n R + B + n Where, R + R + B + n nd thus 1R + B + n. P reds in n trils 1R +B +n 1R +B +n 1R +B +n PR n R + B + n n n 1 p 1 p n f pd p P }{{} Using Fubini s Theorem we cn rewrite this with the summtion inside the integrl. 5 F ρn 1R +B +n n p 1 p n f pd p } {{ } n Let S n be Binomil n, p then PX n n p 1 p n. This gives us some ide of how to rewrite the integrl so we cn show it goes to the bet distribution. Also it is worth noting tht Binomil rndom vrible is the sum of n bernoulli rndom vribles. PS n 1R + B + n f pd p }{{} 1R +B +n PX n

74 Drinne 28 For the next step we will center our probbility within the integrl t nd stndrdize it so we cn use the DeMoivre-Lplce Theorem. PS n 1R + B + n PS n np 1R + B + n np S n np 6 P 1R + B pn PZ + np1 p np1 p np1 p PZ 1 1 p Now we cn define the RHS s X, p p > F ρn X, p f pd p X, p f pd p + X, p f pd p Which is the c.d.f of the bet distribution. p > p < f p f p 5. A RATE FROM BEERY-ESSEEN FOR F ρn F ρ L1 In this section we will derive rte of the convergence for the urn s distribution to the bet distribution using the Beery-Esseen Theorem. F ρn F ρ PS n 1R + B + n f pd p PS n 1R + B + n f pd p f pd p X, p f pd p PS n 1R + B + n X, p f pd p Where X, p is equivlent to the cdf of the Norml so write X, p N [ 1R + B + n]. Also S n is Binomil rndom vrible which is sum of n Bernoulli trils. We wnt to chnge our rndom vrible to wor with Beery-Esseen so we subtrct by np nd divide by σ n. P S n np σ n 1R + B + n np σ n 7 Then, using Berry-Esseen N 1R + B + n np σ f pd p n 3 p f pd p Ψn or our rte function nσ 3 Where p E[ X i np σ n 3 ] E[ X i np 3 1 p 3 p 3 σ ] n σ p + n σ p1 p1 2p 1 p n σ 3 n 3/2 Writing 7 out fully we see tht this is nice integrl to wor with nd the rest follows esily. 3 1 p1 p1 2p 3 1 p R 1 p B+1 d p 1 p R+1 1 p B d p n 2 βr,b σ 3 f pd p n 2 βr,b p1 p 3 p1 p 3 3 n 2 βr,b βr 2,B 1 βr 1,B 2

Rte of Convergence of Poly s Urn to the Bet Distribution 75 3 ΓR + B ΓR 2ΓB 1 n 2 ΓR 1ΓB 2 ΓR ΓB ΓR + B 3 ΓR + B 3 3 n 2 R + B 1R + B 2R + B 3 R 1R 2B 1 R + B 1R + B 2R + B 3 R 1B 1B 2 3 n 2 R + B 1R + B 2R + B 3B R R 1R 2B 1B 2 6. A BETTER RATE USING CRAMÉR S THEOREM From the previous derivtion of the cumultive distribution function we hve: F ρn PX n 1R + B + n f pd p Agin if we notice tht the rndom vrible here is Binomil rndom vrible which is the sum of n Bernoulli rndom vribles we cn rewrite this integrl in similr form tht is of the sme order due to the fct tht the difference between this integrl nd the new integrl is minor s n rnges to infinity. So we hve, F ρn PX n n f pd p Now consider the difference between the previous integrl nd the CDF of the bet distribution. F ρ F ρn f pd p f pd p PS n n f pd p 1 Ps n n f pd p PS n n f pd p PS n n f pd p PS n n f pd p PS n n f pd p P S n n f pd p Notice from Crmérs Theorem tht lim n 1 n logp[s n n] I thus PS n n e ni. Continuing from the previous step we hve the following expression where I sup t R [t logϕt]. 8 PS n n f pd p P S n n f pd p e ni f pd p e ni f pd p Thus ϕt pe t + 1 p nd to get I solve for the supremum. d [ t logpe t + 1 p ] pe t 1 p dt pe t, Solve for t log + 1 p 1 p 1 p 1 p I log log 1 p 1 + p 1 log + 1 log1 + 1log1 p log p

76 Drinne 28 For the second integrl: ϕt pe t + 1 p nd to get I solve for the supremum which turns out to be the sme s tht for I. Writing the full difference of integrls from 8 we hve the following. exp[ n log exp[ n 1 p 1 p log 1 p 1 + 1 p ] f pd p 1 p 1 p log log 1 p 1 + 1 p ] f pd p [ 1 p ] 1 p exp n log log f pd p 1 p 1 e nlog+1 log1 e n 1log1 p log p f pd p e nlog+1 log1 1 p B n 1 1 p R 1 +n 1 βr,b d p e nlog+1 log1 βr + n,b + n1 βr,b 9 e nlog+1 log1 ΓR + nγb + n1 βr,b ΓR + B + n Eqution 9 is good expression to study how the urn goes to the bet distribution. First we will rewrite it in terms of the Sterling formul to pproximte the gmm function. Since this expression is so lrge it will be studied in seprte terms. The following three equtions re pproximtions of the gmm functions in eqution 9. R + n ΓR + n R + n e B + n1 ΓB + n1 B + n1 e R + B + n ΓR + B + n R + B + n e R +n 1 + O 1 n B +n1 1 + O 1 n R +B +n 1 + O 1 n

Rte of Convergence of Poly s Urn to the Bet Distribution 77 Rewriting terms with the pproximtions nd then to bse e yields the eqution we need to bre prt nd study. 9 exp[ nlog+1 log1 +R +nlogr + n+b +n1 logb + n1 R +B +nlogr + B + n]1+o 1 n R + n B + n1 1 The terms of squre roots. L B +n1 R +B +n R +n R + B + n n 2 1 βr,b 1 11 exp[r +nlogr + n+b +n1 logb + n1 R +B +nlogr + B + n] #1 {}}{ exp[r logr + n + B logb + n1 R + B logr + B + n] exp[nlogr + n + n1 logb + n1 nlogr + B + n] }{{} #2 The following is the rewriting of #2. expnlogn + n1 logn1 nlogn B n1 +log 1 + + log 1 + R n log 1 + R + B n n1 n n The expression with the logrithms to powers is fmilir nd is useful to exmine it s limit. B n1 L 1 log 1 + + log 1 + R n log 1 + R + B n B + R 1 n1 n n R + B If we combine wht is left from expression #2 with exp[ nlog + 1 log1 ] from 9 the following results. expnlogn + n1 logn1 nlogn nlog n1 log1 exp[nlogn + n1 logn nlogn] nn n n1 n n 1 Rewriting the eqution 9 with the simplifictions we hve nd lbeling the functions tht we now converge to specific limit we hve. 12 1 βr,b L 1L 1 + O 1 n 1 βr,b n 2 1 + O 1 n

78 Drinne 28 7. CONCLUSION The first rte I derived seems to be incorrect when I exmine numericl exmples of the distnce between the urn s distribution nd the ctul. The second rte does however seem to mtch the numericl evidence. I believe one could derive much better rte if they incorported the reltionship between R nd B into their investigtion. The reson for this is the gret difference between distributions where B R nd the distributions where B R. I would lie to personlly thn the dvisors of the REU which re Nthn Gibson, Yevgeniy Kovchegov, Donld Solmon nd Pul Cull, the orgnizer Dennis Grrity, nd Oregon Stte University. I lso would lie to thn the NSF for their grcious funding. I hd wonderful time nd enjoyed the hospitlity tht I ws privileged to experience. REFERENCES [DR] Durrett, Richrd. Probbility: Theory nd Exmples Second Edition [FW] Feller, Willim. An Introduction to Probbility Theory nd Its Appliction [HF] Hollnder, Frn den. Lrge Devitions Fields Institute Monogrphs [RS] Ross, Sheldon. A First Course in Probbility Seventh Edition

Rte of Convergence of Poly s Urn to the Bet Distribution 79 8. APPENDIX The following is Mthemtic 6 code used to generte the empiricl distributions used for the digrms in this pper. TheoreticlDist[Red_, Blue_, DrwN_] : Module[ {RedInitil, BlueInitil, DepthN}, RedInitil Red; BlueInitil Blue; DepthN DrwN; NewTble {{RedInitil, BlueInitil, 1, 1}}; For [ MinI, MinI < DepthN, MinI++, OldTble NewTble; LoopItertions Length[NewTble NewTble Arry[1 &, {2*Length[OldTble], 4} For [ i 1, i < LoopItertions, i++, *Red* NewTble[[2*i - 1, 1]] OldTble[[i, 1]] + 1; NewTble[[2*i - 1, 2]] OldTble[[i, 2] NewTble[[2*i - 1, 3]] OldTble[[i, 1]]/OldTble[[i, 1]] + OldTble[[i, 2]] // N; NewTble[[2*i - 1, 4]] OldTble[[i, 4]]*NewTble[[2*i - 1, 3] *Blue* NewTble[[2*i, 1]] OldTble[[i, 1] NewTble[[2*i, 2]] OldTble[[i, 2]] + 1; NewTble[[2*i, 3]] OldTble[[i, 2]]/OldTble[[i, 1]] + OldTble[[i, 2]] // N; NewTble[[2*i, 4]] OldTble[[i, 4]]*NewTble[[2*i, 3] *Te NewTble nd Build CDF Arry*

8 Drinne 28 SortedTble SortBy[NewTble, First NewIndex SortedTble[[1, 1] CDFBuildTble Arry[ &, {DepthN + 1, 2} CDFIndex 1; CDFBuildTble[[1, 1]] RedInitil; For [i 1, i < Length[SortedTble], i++, OldIndex NewIndex; NewIndex SortedTble[[i, 1] If [ OldIndex! NewIndex, CDFIndex CDFIndex + 1; CDFBuildTble[[CDFIndex, 1]] NewIndex; CDFBuildTble[[CDFIndex, 2]] SortedTble[[i, 4], CDFBuildTble[[CDFIndex, 2]] CDFBuildTble[[CDFIndex, 2]] + SortedTble[[i, 4] For[ i 1, i < Length[CDFBuildTble], i++, CDFBuildTble[[i, 1]] CDFBuildTble[[i, 1]]/RedInitil + BlueInitil + DepthN // N; If[i > 2, CDFBuildTble[[i, 2]] CDFBuildTble[[i, 2]] + CDFBuildTble[[i - 1, 2] CDFBuildTbleFinl Arry[ &, {2*Length[CDFBuildTble] + 2, 2} For[ i 1, i < Length[CDFBuildTble] - 1, i++, If[i > 1, CDFBuildTbleFinl[[2*i + 1, 1]] CDFBuildTble[[i, 1] CDFBuildTbleFinl[[2*i + 1, 2]] CDFBuildTble[[i, 2] CDFBuildTbleFinl[[2*i + 2, 1]] CDFBuildTble[[i + 1, 1]

Rte of Convergence of Poly s Urn to the Bet Distribution 81 CDFBuildTbleFinl[[2*i + 2, 2]] CDFBuildTble[[i, 2] CDFBuildTbleFinl[[2, 1]] CDFBuildTble[[1, 1] CDFBuildTbleFinl[[Length[CDFBuildTbleFinl] - 1, 1]] CDFBuildTble[[Length[CDFBuildTble], 1] CDFBuildTbleFinl[[Length[CDFBuildTbleFinl] - 1, 2]] 1; CDFBuildTbleFinl[[Length[CDFBuildTbleFinl], 1]] 1; CDFBuildTbleFinl[[Length[CDFBuildTbleFinl], 2]] 1; *Distnce Stuff* MxAbDistnce ; PositionAbDistnce ; DistFunction CDFBuildTbleFinl; For[i 1, i < Length[CDFBuildTbleFinl], i++, AbDistnce Abs[CDFBuildTbleFinl[[i, 2]] - CDF[BetDistribution[RedInitil, BlueInitil], CDFBuildTbleFinl[[i, 1]]] // N DistFunction[[i, 2]] AbDistnce; If[AbDistnce > MxAbDistnce, MxAbDistnce AbDistnce; PositionAbDistnce DistFunction[[i, 1] The next section of code will disply the plot of the empiricl distribution vs. the ctul bet distribution. First execute the pervious section nd then this section. Mnipulte[TheoreticlDist[R, B, n Show[ListLinePlot[CDFBuildTbleFinl, PlotRnge -> {{, 1}, {, 1}}, PlotStyle -> Directive[Thicness[Lrge], Red]], Plot[CDF[BetDistribution[R, B], x], {x,, 1}, PlotStyle -> Directive[Thicness[Lrge], Blue]]], {R, 1, 1, 1}, {B, 1, 1, 1}, {n, 1, 15, 1}] WINONA STATE UNIVERSITY E-mil ddress: jdrinne4878@winon.edu