COUNTS OF FAILURE STRINGS IN CERTAIN BERNOULLI SEQUENCES

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1 COUNTS OF FAILURE STRINGS IN CERTAIN BERNOULLI SEQUENCES LARS HOLST Deprtment of Mthemtics, Royl Institute of Technology SE Stockholm, Sween E-mil: October 6, 2006 Abstrct In sequence of inepenent Bernoulli trils the probbility for success in the k:th tril is p k, k = 1, 2,.... The number of strings with given number of filures between two subsequent successes is stuie. Explicit expressions for istributions n moments re obtine for p k = / + b + k 1, > 0, b 0. Also the limit behviour of the longest filure string in the first n trils is consiere. For b = 0 the strings correspon to cycles in rnom permuttions. Keywors: Binomil moments; Ewens Smpling Formul; Hoppe s urn; Poisson istribution; Poisson Dirichlet istribution; Póly s urn; rnom permuttions; recors; spcings; sums of inictors ms 2000 subject clssifiction: primry 60c05 seconry 60k99 Running title: Counts of filure strings 1 Introuction In n infinite sequence of inepenent Bernoulli trils the probbility for success in the k:th tril is p k for k = 1, 2,.... A string is string SF... F S of 1 filures between two subsequent successes. We will stuy the number of such strings. Explicit results re obtine for p k = /+b+k 1, > 0, b 0. To our knowlege only specil cses hve been stuie previously.

2 For = 1, b = 0, tht is p k = 1/k, 1 strings correspon to ouble recors in recor sequence. Hhlin 1995 prove tht the totl number of such recors is Po1 Poisson istribute with men 1. After tht, n unpublishe proof by Diconis inspire number of stuies on 1 strings, see Chern et l 2000, Mori 2001, Joffe et l 2004 n the references therein. Sethurmn n Sethurmn 2004 stuie strings for = 1, b > 0, n obtine the joint istribution of the number of strings for = 1, 2,.... For > 0 n b = 0, strings re closely connecte with cycle lengths in rnom permuttions, see e.g. Arrti, Brbour n Tvré 2003 pge 95. In Section 2 we introuce nottions n erive recursions for the binomil moments of the number of strings in finite sequence for generl p k s. The specil cse p k = / + k 1, connecte with rnom permuttions, is stuie in Section 3. In Section 4 we erive the joint istribution of the totl number of strings, = 1, 2,..., n stuy the limit behviour of the longest filure string in the first n trils in n infinite Bernoulli sequence with p k = / + b + k 1. 2 Generl cse: nottions n moments In the following I 1, I 2,... is sequence of inepenent Bernoulli rnom vribles, I k is Bep k, tht is P I k = 1 = 1 P I k = 0 = p k. The number of strings in the first n trils is with men n M n = I k 1 I k+1 1 I k+ 1 I k+ n EM n = p k 1 p k+1 1 p k+ 1 p k+. Note tht M n = 0 for n n n 1 j=1 jm jn n 1. Implicitly, the following result gives the istribution of M 1n,..., M n. Proposition 2.1 For the binomil moments M 1n f n,..., r = E Mn with n 1 n j=1 jr j n 1, the recursion hols: r 2

3 f n+1,..., r = f n,..., r +p n+1 fn 1, r 2,..., r 1 p n f n 1 1, r 2,..., r ] +p n+1 1 p n f n 1, r 2 1, r 3,... 1 p n 1 f n 2, r 2 1, r 3,... ] p n+1 1 p n 1 p n +2 f n +1,..., r 1, r 1 1 p n +1 f n,..., r 1, r 1 ]. Proof. Using generting functions n the inepenence between the I k s we get ] E t M 1,n+1 1 t M,n+1 = E t M 1n 1 t M n 1 + t1 1I n I n t2 1I n 1 1 I n I n+1 ] = E t M 1n 1 t M n 1 + t1 1I n I n+1 + t 2 1I n 1 1 I n I n ] ] = E t M 1n 1 t M n + t 1 1p n+1 E t M 1n 1 t M n 1 1 In ] +t 2 1p n+1 E t M 1n 1 t M n 1 1 In 1 ] 1 I n +... ] = E t M 1n 1 t M n ]+t 1 1p n+1 E t M 1n 1 t M n 1 p n E t M 1,n 1 1 t M,n 1 ] +t 2 1p n+1 1 p n E t M 1,n 1 1 t M,n 1 1 p n 1 E t M 1,n 2 1 t M,n Expnsion in series roun t 1 = 1,..., t = 1 proves the ssertion. Incluing the string SF F with 1 filures fter the lst success in the count we get the rnom vrible N n = M n + I n +1 1 I n +2 1 I n with men EN n = EM n + p n +1 1 p n +2 1 p n. Proposition 2.2 For p n+1 > 0 it hols for the binomil moments: N 1n Nn M 1n Mn E = E + 1 M 1,n+1 E p n+1 r M,n+1 r r M 1n E Mn r ]. 3

4 Proof. By the lw of totl probbility we hve E t M 1,n+1 1 t M,n+1 = p n+1 E t N 1n 1 t N n + 1 p n+1 E t M 1n 1 t M n, from which the ssertion follows by expnsion in series. In n infinite sequence the totl number of strings M = I k 1 I k+1 1 I k+ 1 I k+ < + with probbility one if n only if p k 1 p k+1 1 p k+ 1 p k+ < +. Inee, by splitting the series for M into + 1 inepenent series this follows from the Borel-Cntelli lemms, cf. Mori 2001 pge The cse p k = / + k 1 Following Knuth 1992 we enote escening n scening fctorils by x n = xx 1 x n + 1, x n = xx + 1 x + n 1 = n n k ] x k, where ] n k is cycle number or signless Stirling number of the first kin. We ssume in the rest of this section tht p k = /+k 1 with > 0. Close simple formuls cn be obtine for the binomil moments. Note tht n 1 j=1 jm jn n 1 n n j=1 jn jn = n with probbility one. Proposition 3.1 For m = j=1 jr j n 1: M 1n E Mn r Proof. By telescoping sums we get EM n = n = n 1m + n 1 m k 1 + k 4 /j r j. r j! j=1 + k k + 1

5 = n 1 ] k k 1 + k = n + n 1 = n 1 + n 1. + k + 2 Hence the ssertion hols for EM n. By elementry clcultions the recursion in Proposition 2.1 is verifie. The proof is finishe by inuction. Proposition 3.2 For n n m = j=1 jr j n: n for n j=1 jx j = n: N 1n E Nn r = n m + n 1 m P N 1n = x 1,..., N nn = x n = n! n /j r j, r j! j=1 n /j x j. x j! Proof. Using Propositions 2.2 n 3.1 the first ssertion follows from n elementry clcultion. We hve using generting functions tht E t N 1n 1 t N nn n = E 1 + t1 1 N 1n 1 + t n 1 N nn = N 1n Nnn E t 1 1 r1 t n 1 r n = N 1n E Nnn r n r n j=1 1 x1 1 r n x n r1 x 1 rn x n t x 1 1 tx n n. As n 1 jn jn = n the binomil moments isppers for n 1 jr j > n. Therefore for n 1 jx j = n we hve r j = x j in the summtion. Thus N 1n P N 1n = x 1,..., N nn = x n = E proving the secon ssertion. x 1 Nnn The istribution of N 1n,..., N nn is the fme Ewens Smpling Formul. Furthermore, N n is the number of strings in 1I 2 I 3... I n 1. Using this, the lst proposition cn be erive by combintoril rguments, cf. Arrti et l 2003 pge 95. In tht context N n is interprete s the number of cycles of length in rnom 5 x n,

6 permuttion of 1, 2,..., n bise by Kn, where K n = n I k is the number of cycles with the istribution The moment convergence n ] j P K n = j =, j = 1, 2,..., n. j n M 1n E Mn r /j r j, n, r j! implies the following result, well known for bise rnom permuttions, see Arrti et l 2003 pge 96. Proposition 3.3 The number of strings M 1, M 2,... re inepenent Poisson rnom vribles with EM = /. j=1 4 The cse p k = / + b + k 1 In this section we ssume tht p k = / + b + k 1 with > 0 n b > 0. Clerly M = I k 1 I k+1 1 I k+ 1 I k+ < + with probbility one. Mori 2001 erive the istribution of M 1. For the specil cse = 1 Sethurmn n Sethurmn 2004 obtine the joint istribution of M 1, M 2,.... Using ifferent methos we generlize their result to ny > 0. Let U be Bet, b, tht is rnom vrible with ensity f U u = Γ + b ΓΓb u 1 1 u b 1, 0 < u < 1. Theorem 4.1 Conitionl on Bet,b rnom vrible U, the number of strings M 1, M 2,... re inepenent Poisson rnom vribles with EM U = 1 1 U, = 1, 2,.... Proof. We introuce the following mixture of Póly s n Hoppe s urn moels. An urn contins initilly one white n one blck bll of weights n b respectively. Blls re rwn t rnom proportionl to weights. The white n the blck bll re replce together with new bll of colour not present in the urn, other blls 6

7 re replce together with one new bll of the sme colour. All new blls hve weight one. Obviously, the probbility of rwing the white bll t rwing k is p k = / + b + k 1. Generte sequence of W s n B s. We get W if rwing the white bll or bll of colour emnting from rw of the white, else we get B. This sequence is s rwing from n orinry Póly urn. Note tht the sequence is exchngeble. Therefore, by e Finetti s theorem the sequence cn be thought of s hving been generte by first observing the Bet, b rnom vrible U n then, conitionl on the outcome U = u, generting sequence of inepenent Beu rnom vribles, with 1 corresponing to W n 0 to B. In the subsequence of W s in the originl sequence I 1, I 2,... the probbility of getting the white bll t the j:th tril is p j = / + j 1. Accoring to Proposition 3.3 the number of strings in the subsequence, M, is Po/ n M1, M 2,... re inepenent. Recll the following well known fct. If the rnom vrible ξ is Poµ n inepenent of the inepenent Bep rnom vribles ε 1, ε 2,..., then ξ j=1 ε j n ξ j=1 1 ε j re inepenent Poµp n Poµ1 p respectively. Consier the 1 strings in the subsequence of W s. Ech such 1 string is lso 1 string in the originl sequence, provie it ws not interrupte by B. The probbility for interruption is 1 u. As M1 is Po it follows from the fct bove n the inepenence, tht conitionl on U = u, the totl number of 1 strings in the originl sequence, M 1, is Pou n inepenent of M1 M 1 which is Po1 u. For the 2 strings we cn rgue in similr wy s bove. Conitionl on U = u, the rnom vrible M 2 is Poisson with men 2 u2 + 1 uu = u2 n inepenent of M 1. The rgument extens, M is Poisson with conitionl men u + 1 u 1 1 u + 1 u 2 1 u u1 u = 1 1 u n inepenent of M 1, M 2,..., M 1,. 7

8 Finlly, consier long strings of filures. Let the lst success in the first n trils occur t tril n+1 A 1n ; if there is no success set A 1n = 0. We hve for j = 1, 2,..., n P A 1n > j = b + n j + b + n 1 b + n jj Γb + n = = + b + n j j Γb + n j Γ + b + n j. Γ + b + n For j, n such tht j/n x, 0 < x < 1, Stirling s formul gives P A 1n /n > j/n 1 x, n, tht is A 1n /n converges in istribution to Bet1,. In similr wy we fin for the number of trils between the lst n the secon lst success, A 2n, tht A 1n, A 2n /n U 1, 1 U 1 U 2, n, in istribution, where U 1, U 2 re inepenent Bet1, rnom vribles. The proceure cn be repete in like mnner. The limit behviour of the long strings is s if A 1n, A 2n,... h been cycle lengths in n bise rnom permuttion, see Arrti et l 2003 Section 5.4. The limit istribution of the size orere A s is the Poisson Dirichlet istribution with prmeter. In prticulr we hve: Theorem 4.2 For the longest string of filures in the first n trils: mxa 1n, A 2n,... /n L 1 = mxu 1, 1 U 1 U 2, 1 U 1 1 U 2 U 3,..., n, in istribution, where U 1, U 2,... re inepenent Bet1, rnom vribles. Vrious formuls connecte with the rnom vrible L 1 cn be foun in Arrti et l 2003 Section

9 References 1] Arrti, R., Brbour, A.D. n Tvré, S Logrithmic Combintoril Structures: Probbilistic Approch. Europen Mthemticl Society Publishing House, ETH-Zentrum, Zürich. 2] Chern, H.-H., Hwng, H.-K. n Yeh, Y.-N Distribution of the number of consecutive recors. Rnom Structures & Algorithms, ] Hhlin, L.O Double Recors. Uppsl University Deprtment of Mthemtics Report, 1995:12. Licentit thesis. 4] Joffe, A., Mrchn, E., Perron, F. n Popiuk, P On sums of proucts of Bernoulli vribles n rnom permuttions. Journl of Theoreticl Probbility, ] Knuth, D Two notes on nottions. The Americn Mthemticl Monthly, ] Mori, T.F On the istribution of sums of overlpping proucts. Act Scientirum Mthemtic Szege, ] Sethurmn, J. n Sethurmn, S On counts of Bernoulli strings n connections to rnk orers n rnom permuttions. In A festschrift for Hermn Rubin. IMS Lecture Notes Monogrph Series, , Institute of Mthemticl Sttistics, Bechwoo, Ohio. 9