A study of counts of Bernoulli strings via conditional Poisson processes
Authors:
Fred W. Huffer, Jayaram Sethuraman and Sunder Sethuraman
Journal:
Proc. Amer. Math. Soc. 137 (2009), 21252134
MSC (2000):
Primary 60C05; Secondary 60K99
Published electronically:
December 30, 2008
MathSciNet review:
2480294
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A sequence of random variables, each taking values 0 or , is called a Bernoulli sequence. We say that a string of length occurs in a Bernoulli sequence if a success is followed by exactly failures before the next success. The counts of such strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic 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 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.
 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. MR 1177897
(94a:60003)
 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. MR 2032426
(2004m:60004)
 3.
Richard
Arratia and Simon
Tavaré, The cycle structure of random permutations,
Ann. Probab. 20 (1992), no. 3, 1567–1591. MR 1175278
(93g:60013)
 4.
HuaHuai
Chern, HsienKuei
Hwang, and YeongNan
Yeh, Distribution of the number of consecutive records,
Proceedings of the Ninth International Conference “Random Structures
and Algorithms” (Poznan, 1999), 2000, pp. 169–196. MR 1801131
(2002c:60006), http://dx.doi.org/10.1002/10982418(200010/12)17:3/4<169::AIDRSA1>3.0.CO;2K
 5.
W.
Feller, The fundamental limit theorems in
probability, Bull. Amer. Math. Soc. 51 (1945), 800–832. MR 0013252
(7,128i), http://dx.doi.org/10.1090/S000299041945084481
 6.
J.
K. Ghosh and R.
V. Ramamoorthi, Bayesian nonparametrics, Springer Series in
Statistics, SpringerVerlag, New York, 2003. MR 1992245
(2004g:62004)
 7.
Lars
Holst, Counts of failure strings in certain Bernoulli
sequences, J. Appl. Probab. 44 (2007), no. 3,
824–830. MR 2355594
(2008i:60014), http://dx.doi.org/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. MR 2054589
(2005e:60023), http://dx.doi.org/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 7490.
 10.
Ramesh
M. Korwar and Myles
Hollander, Contributions to the theory of Dirichlet processes,
Ann. Probability 1 (1973), 705–711. MR 0350950
(50 #3442)
 11.
Tamás
F. Móri, On the distribution of sums of overlapping
products, Acta Sci. Math. (Szeged) 67 (2001),
no. 34, 833–841. MR 1876470
(2002h:60024)
 12.
Sidney
Resnick, Adventures in stochastic processes, Birkhäuser
Boston, Inc., Boston, MA, 1992. MR 1181423
(93m:60004)
 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. MR 2126893
(2006d:60020), http://dx.doi.org/10.1214/lnms/1196285386
 1.
 Arratia, R., Barbour, A.D., and Tavaré, S. (1992), Poisson process approximations for the Ewens sampling formula. Ann. Appl. Probab. 2 519535. MR 1177897 (94a:60003)
 2.
 Arratia, R., Barbour, A.D., and Tavaré, S. (2003), Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society, Zürich. MR 2032426 (2004m:60004)
 3.
 Arratia, R., and Tavaré, S. (1992), The cycle structure of random permutations. Ann. Probab. 20 15671591. MR 1175278 (93g:60013)
 4.
 Chern, H.H., Hwang, H.K., and Yeh, Y.N. (2000), Distribution of the number of consecutive records. Random Structures and Algorithms 17 169196. MR 1801131 (2002c:60006)
 5.
 Feller, W. (1945), The fundamental limit theorems in probability. Bull. Amer. Math. Soc. 51 800832. MR 0013252 (7:128i)
 6.
 Ghosh, J.K., and Ramamoorthi, R.V. (2003), Bayesian Nonparametrics, SpringerVerlag, New York. MR 1992245 (2004g:62004)
 7.
 Holst, Lars (2007), Counts of failure strings in certain Bernoulli sequences. J. Appl. Probab. 44 824830. MR 2355594 (2008i:60014)
 8.
 Joffe, A., Marchand, E., Perron, F., and Popadiuk, P. (2004), On sums of products of Bernoulli variables and random permutations. Journal of Theoretical Probability 17 285292. MR 2054589 (2005e:60023)
 9.
 Kolchin, V.F. (1971), A problem of the allocation of particles in cells and cycles of random permutations. Theory Probab. Appl. 16 7490.
 10.
 Korwar, R.M., and Hollander, M. (1973), Contributions to the theory of Dirichlet processes. Ann. Probab. 1 705711. MR 0350950 (50:3442)
 11.
 Móri, T.F. (2001), On the distribution of sums of overlapping products. Acta Scientarium Mathematica (Szeged) 67 833841. MR 1876470 (2002h:60024)
 12.
 Resnick, S.I. (1994), Adventures in Stochastic Processes. Second Ed., Birkhäuser, Boston. MR 1181423 (93m:60004)
 13.
 Sethuraman, Jayaram, and Sethuraman, Sunder (2004), On counts of Bernoulli strings and connections to rank orders and random permutations. In A festschrift for Herman Rubin. IMS Lecture Notes Monograph Series 45 140152. MR 2126893 (2006d:60020)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
60C05,
60K99
Retrieve articles in all journals
with MSC (2000):
60C05,
60K99
Additional Information
Fred W. Huffer
Affiliation:
Department of Statistics, Florida State University, Tallahassee, Florida 32306
Email:
huffer@stat.fsu.edu
Jayaram Sethuraman
Affiliation:
Department of Statistics, Florida State University, Tallahassee, Florida 32306
Email:
sethu@stat.fsu.edu
Sunder Sethuraman
Affiliation:
Department of Mathematics, 396 Carver Hall, Iowa State University, Ames, Iowa 50011
Email:
sethuram@iastate.edu
DOI:
http://dx.doi.org/10.1090/S0002993908097931
PII:
S 00029939(08)097931
Keywords:
Bernoulli,
cycles,
strings,
spacings,
nonhomogeneous,
Poisson processes,
random permutations
Received by editor(s):
January 14, 2008
Received by editor(s) in revised form:
September 25, 2008
Published electronically:
December 30, 2008
Additional Notes:
This research was partially supported by AROW911NF0410333, NSAH982300510041, and NSFDMS0504193.
Communicated by:
Edward C. Waymire
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
