TH NUMBR OF TWO CONSCUTIV SUCCSSS IN A HOPP-PÓLYA URN LARS HOLST Depatment of Mathematics, Royal Institute of Technology S 100 44 Stocholm, Sweden -mail: lholst@math.th.se Novembe 27, 2007 Abstact In a sequence of independent Benoulli tials the pobability of success in the :th tial is p a/a+b+ 1. An explicit fomula fo the binomial moments of the numbe of two consecutive successes in the fist n tials is obtained and some consequences of it ae deived. Keywods: Benoulli tials; binomial moments; Hoppe s un; Ovelapping indicatos; Pólya s un; Recods; Random pemutations 2000 Mathematics Subject Classification: Pimay 60C05 Seconday 60K99 1 Intoduction An un contains initially one white and one blac ball of weight a > 0 and b 0 espectively. Balls ae andomly dawn fom the un with pobabilities popotional to weights. vey time the white o the blac ball is dawn it is eplaced into the un togethe with a ball of weight one and with a colou not aleady in the un, else a ball is eplaced togethe with a copy of it. We call this dawing scheme a Hoppe-Pólya un. If b 0 thee is no blac ball, the so called Hoppe s un. If all balls emanating fom a daw of the white blac ae coloued white blac we get the well nown Pólya s un.
Let the sequence of independent Benoulli andom vaiables I 1, I 2, I 3,... indicate the dawings of the white ball, the successes o ecods in the Hoppe-Pólya un. Obviously, p P I 1 1 P I 0 a/a + b + 1, 1, 2,.... The numbe of successes in the fist n tials can be witten K n I 1 + I 2 + + I n, and the numbe of two consecutive successes is M n I 1 I 2 + I 2 I 3 + + I n 1 I n. An explicit fomula fo the binomial moments of M n is the main esult of this pape. Note that 0 M n n 1. Fo p a/a + b + 1 the Boel-Cantelli Lemma implies that M I I +1 < + with pobability one. Fo the case a 1 and b 0, that is p 1/, connected with ecod values and andom pemutations, Hahlin 1995 poved that M is Poisson distibuted with mean 1. Afte that an unpublished poof of the same esult by Diaconis inspied a numbe of studies on the distibution of M, see Chen et al. 2000, Moi 2001, Joffe et al. 2004, Sethuaman and Sethuaman 2004, Holst 2007, and the efeences theein. To ou nowledge the esult in this pape on the distibution of M n fo finite n has not been obtained peviously. 2 Notations and facts Following Knuth 1992 we denote falling and ising factoials by x n xx 1 x n + 1, x n xx + 1 x + n 1 n j1 [ n j ] x j, whee [ n j ] is a cycle numbe o signless Stiling numbe of the fist ind. Recall the combinatoial intepetation: [ n j ] is the numbe of pemutations of 1, 2,..., n with j cycles. Fo K n equals the numbe of successes in the n fist tials, we have n x K n a a + b + 1 x + 1 a ax + bn a + b + 1 a + b n 2
n [ n ] ax + b j n j a + b n j1 Hence fo i 0, 1, 2,... n P K n i n ji i0 x i [ n j ] a + b j a + b n n [ n j ji ] j a i b j i i a + b n. j a i b j i. i a + b a + b In paticula fo b 0, that is Hoppe s un, we get the cycle distibution [ n ] a i P K n i, i 1, 2,..., n, i an fo an a-biased andom pemutations; see Aatia et al. 2003 page 100. The numbe of times the white ball o balls emanating fom it has been dawn in the n fist tials, X n, has the Pólya-ggenbege distibution P X n i n i a i b n i a + b n n i whee U is a Betaa, b andom vaiable with density f U u U i 1 U n i, i 0, 1, 2,..., n, Γa + b ΓaΓb ua 1 1 u b 1, 0 < u < 1. Using the binomial distibution we get fo 1, 2,..., n n Xn i n U i 1 U n i i i n U n a a + b. Recall that a andom vaiable S with the hypegeometic distibution c d P S i i n i has the binomial moment S 0 c+d n n c c + d. Fo an intege-valued andom vaiable Z 0 having pobability geneating function with adius of convegence lage than 1 we have Z Z x Z 1 + x 1 Z x 1 x i 1 i, i 3 i0 i
which gives the pobability function of Z expessed in binomial moments P Z i 1 i i i Note that if 0 Z < n then Z 0 fo n. Z, i 0, 1, 2,.... 3 The numbe of two consecutive successes The following esult gives implicitely the distibution of M n. Theoem 3.1 Fo p a/a + b + 1 and 1, 2,..., n 1: a a + b + n 1 1 n a a + b. Befoe poving the theoem we conside the special case b 0, that is Hoppe s un. A moe geneal esult is Poposition 3 in Holst 2007. Lemma 3.1 Fo p a/a + 1 and 1, 2,..., n 1: n 1 a a + n 1. Poof. Fo N n M n + I n we have which implies Nn+1 t N n+1 p n+1 t N n t + 1 p n+1 t M n, p n+1 Nn Nn + + 1 p n+1 1 Fo p a/a + 1 the andom vaiable N n has the same distibution as the numbe of fix-points in an a-biased andom pemutation of 1, 2,..., n, and Nn n a a + n 1, see Aatia et al. 2003 pages 95 and 96. Using this and the elation above poves the assetion.. 4
Poof of the theoem. Conside the Hoppe-Pólya un and the andom vaiable X n intoduced in Section 2. In the X n white dawings the pobability of getting the white ball in the j:th tial is p j a/a + j 1. Given X n x, the numbe of times the white ball was consecutively dawn in these white dawings, Mx, is distibuted as in the lemma. Conditional on X n x we can ague as follows. Let among the x white daws W 1 denote a dawing giving the white ball and W 0 giving a ball emanating fom the white. B denotes a blac dawing. The esult of the white daws can be witten W 1 W i2 W i3... W ix whee i 2,..., i x ae 0 o 1. Fo Mx y we have that y of the pais W 1 W i2,..., W ix 1W ix ae of type W 1 W 1. Fo M n z consecutive daws W 1 W 1 among the oiginal n daws with x W s and n x B s, thee ae z pais of the y W 1 W 1 -pais among the white daws which ae intact, and y z which ae split by at least one B between W 1 W 1. The numbe of ways to choose the pais to be intact is y z. Afte such a splitting thee ae x z fee W s to combine with n x y z fee B s and thee ae n y x z such combinations. As each combination of x W s and n x B s has the same pobability 1/ n x we get P M n z X n x y n y P Mx y z x z n. y x Thus M n :s pobability function can be witten P M n z y n y P X n xp Mx y z x z n, x,y x with the binomial moment x,y P X n xp M x y z y n y z z x z n. x Using the fomula fo the binomial moment of the hypegeometic distibution and the lemma we get x y P X n xp Mx y n x,y x x x n x n P X n x y n a x b n x x a + b n 5 y P Mx y x 1 a a + x 1.
Hence the binomial moment of the Pólya-ggenbege distibution gives a a + b + n 1 a a + b + n 1 which poves the assetion. x a a + b + n 1 a a + b + n 1 n a x b n x x a + b n t 1 x 1 t n a t b n t t a + b n 1 Xn 1 n a a + b, The distibution of M is obtained in Moi 2001. It is a special case of the distibution in Theoem 1 in Holst 2007. Coollay 3.1 Conditional on a Betaa,b andom vaiable U, M is Poisson distibuted with mean au. Poof. Fom the theoem it follows that a a! a + b, n. As U a /a + b we get using the Poisson distibution that M M U au /! a U /! a! The assetion follows fom the moment convegence. a a + b. The distibution of M n fo p p is studied in Hiano et al. 1991 and the efeences theein. Letting a, b such that a/a + b p we obtain thei esult. Coollay 3.2 Fo p p and 1, 2,..., n 1: 1 p n p. 6
Finally conside the Pólya un stating with one white ball of weight a and one blac ball of weight b. vey dawn ball is eplaced togethe with one ball of the same colou and of weight one. In n dawings the numbe of times a white ball is dawn, X n, has the Pólya-ggenbege distibution. Let Y n be the numbe of times a white ball is consecutively dawn. Coollay 3.3 Fo the Pólya un and 1, 2,..., n 1: Yn 1 n a + a + b. + Poof. Set J 1, if the :th dawn ball is white, else J 0. It is a well nown easily poved fact that conditional on a Betaa, b andom vaiable U the andom vaiables J 1, J 2,... ae independent and Benoulli distibuted with success pobability U. Thus it follows fom the pevious coollay that Yn U which poves the assetion. Refeences 1 n U 1 n U +, [1] Aatia, R., Babou, A.D. and Tavaé, S. 2003. Logaithmic Combinatoial Stuctues: a Pobabilistic Appoach. uopean Mathematical Society Publishing House, TH-Zentum, Züich. [2] Chen, H.-H., Hwang, H.-K. and Yeh, Y.-N. 2000. Distibution of the numbe of consecutive ecods. Random Stuctues Algoithms 17, 169 196. [3] Hahlin, L.O. 1995. Double Recods. Res. Rep. 1995:12, Depatment of Mathematics, Uppsala Univesity. [4] Hiano, K., Ai, S., Kashiwagi, N. and Kuboi, H. 1991. On Ling s binomial and negative binomial distibutions of ode. Statistics Pobability Lettes 11, 503 509. [5] Holst, L. 2007. Counts of failue stings in cetain Benoulli sequences. J. Appl. Pob. 44, 824 830. [6] Joffe, A., Machand,., Peon, F. and Popadiu, P. 2004. On sums of poducts of Benoulli vaiables and andom pemutations. J. Theoet. Pob. 17, 285 292. 7
[7] Knuth, D. 1992. Two notes on notations. Ame. Math. Monthly 99, 403 422. [8] Moi, T.F. 2001. On the distibution of sums of ovelapping poducts. Acta Scientiaum Mathematica Szeged 67, 833 841. [9] Sethuaman, J. and Sethuaman, S. 2004. On counts of Benoulli stings and connections to an odes and andom pemutations. In A festschift fo Heman Rubin IMS Lectue Notes Monog. Se. 45, Institute of Mathematical Statistics, Beachwood, Ohio, pp. 140 152. 8