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A study of counts of Bernoulli strings via conditional Poisson processes. (English) Zbl 1165.60305

Summary: A sequence of random variables, each taking values 0 or 1, is called a Bernoulli sequence. We say that a string of length \( d\) occurs in a Bernoulli sequence if a success is followed by exactly \( (d-1)\) failures before the next success. The counts of such \( d\)-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic \( d\)-cycle counts in random permutations.
In this paper, we give a new framework, in terms of conditional Poisson processes, which allows for a quick characterization of the joint distribution of the counts of all \( d\)-strings, in a general class of Bernoulli sequences, as certain mixtures of the product of Poisson measures. In particular, this general class includes all Bernoulli sequences considered in the literature, as well as a host of new sequences.

MSC:

60C05 Combinatorial probability
60K99 Special processes
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[1] Richard Arratia, A. D. Barbour, and Simon Tavaré, Poisson process approximations for the Ewens sampling formula, Ann. Appl. Probab. 2 (1992), no. 3, 519 – 535. · Zbl 0756.60006
[2] Richard Arratia, A. D. Barbour, and Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2003. · Zbl 1040.60001
[3] Richard Arratia and Simon Tavaré, The cycle structure of random permutations, Ann. Probab. 20 (1992), no. 3, 1567 – 1591. · Zbl 0759.60007
[4] Hua-Huai Chern, Hsien-Kuei Hwang, and Yeong-Nan Yeh, Distribution of the number of consecutive records, Proceedings of the Ninth International Conference ”Random Structures and Algorithms” (Poznan, 1999), 2000, pp. 169 – 196. , https://doi.org/10.1002/1098-2418(200010/12)17:3/43.0.CO;2-K · Zbl 0969.60017
[5] W. Feller, The fundamental limit theorems in probability, Bull. Amer. Math. Soc. 51 (1945), 800 – 832. · Zbl 0060.28702
[6] J. K. Ghosh and R. V. Ramamoorthi, Bayesian nonparametrics, Springer Series in Statistics, Springer-Verlag, New York, 2003. · Zbl 1029.62004
[7] Lars Holst, Counts of failure strings in certain Bernoulli sequences, J. Appl. Probab. 44 (2007), no. 3, 824 – 830. · Zbl 1132.60011 · doi:10.1239/jap/1189717547
[8] Anatole Joffe, Éric Marchand, François Perron, and Paul Popadiuk, On sums of products of Bernoulli variables and random permutations, J. Theoret. Probab. 17 (2004), no. 1, 285 – 292. · Zbl 1054.60013 · doi:10.1023/B:JOTP.0000020485.34082.8c
[9] Kolchin, V.F. (1971), A problem of the allocation of particles in cells and cycles of random permutations. Theory Probab. Appl. 16 74-90. · Zbl 0239.60014
[10] Ramesh M. Korwar and Myles Hollander, Contributions to the theory of Dirichlet processes, Ann. Probability 1 (1973), 705 – 711. · Zbl 0264.60084
[11] Tamás F. Móri, On the distribution of sums of overlapping products, Acta Sci. Math. (Szeged) 67 (2001), no. 3-4, 833 – 841. · Zbl 1001.60015
[12] Sidney Resnick, Adventures in stochastic processes, Birkhäuser Boston, Inc., Boston, MA, 1992. · Zbl 0762.60002
[13] Jayaram Sethuraman and Sunder Sethuraman, On counts of Bernoulli strings and connections to rank orders and random permutations, A festschrift for Herman Rubin, IMS Lecture Notes Monogr. Ser., vol. 45, Inst. Math. Statist., Beachwood, OH, 2004, pp. 140 – 152. · Zbl 1268.60011 · doi:10.1214/lnms/1196285386
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