## Electronic Journal of Probability

### Random Discrete Distributions Derived from Self-Similar Random Sets

#### Abstract

A model is proposed for a decreasing sequence of random variables $(V_1, V_2, \cdots)$ with $\sum_n V_n = 1$, which generalizes the Poisson-Dirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let $V_n$ be the length of the $n$th longest component interval of $[0,1]\backslash Z$, where $Z$ is an a.s. non-empty random closed of $(0,\infty)$ of Lebesgue measure $0$, and $Z$ is self-similar, i.e. $cZ$ has the same distribution as $Z$ for every $c > 0$. Then for $0 \le a < b \le 1$ the expected number of $n$'s such that $V_n \in (a,b)$ equals $\int_a^b v^{-1} F(dv)$ where the structural distribution $F$ is identical to the distribution of $1 - \sup ( Z \cap [0,1] )$. Then $F(dv) = f(v)dv$ where $(1-v) f(v)$ is a decreasing function of $v$, and every such probability distribution $F$ on $[0,1]$ can arise from this construction.

#### Article information

Source
Electron. J. Probab., Volume 1 (1996), paper no. 4, 28 pp.

Dates
Accepted: 20 February 1996
First available in Project Euclid: 25 January 2016

https://projecteuclid.org/euclid.ejp/1453756467

Digital Object Identifier
doi:10.1214/EJP.v1-4

Mathematical Reviews number (MathSciNet)
MR1386296

Zentralblatt MATH identifier
0891.60042

Subjects
Primary: 60G18: Self-similar processes
Secondary: 60G57: Random measures 60K05: Renewal theory

Rights

#### Citation

Pitman, Jim; Yor, Marc. Random Discrete Distributions Derived from Self-Similar Random Sets. Electron. J. Probab. 1 (1996), paper no. 4, 28 pp. doi:10.1214/EJP.v1-4. https://projecteuclid.org/euclid.ejp/1453756467

#### References

• C. Antoniak, Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems, Ann. Statist. 2, (1974) 1152–1174.
• J. Azema, Sur les fermes aleatoires. In Seminaire des Probabilites XIX, Lectures Notes in Math. 1123 Springer, (1985) 397–495.
• H. Berbee, On covering single points by randomly ordered intervals, Ann. Probability, 9, (1981) 520–528.
• P.J. Burville and J.F.C. Kingman, On a model for storage and search, J. Appl. Probab., 10, (1973) 697–701.
• P. Carmona, F. Petit and M. Yor, Some extensions of the arc sine law as partial consequences of the scaling property of Brownian motion, Probab. Th. Rel. Fields, 100, (1994) 1–29.
• P. Donnelly, Partition structures, Polya urns, the Ewens sampling formula, and the ages of alleles, Theoretical Population Biology, 30, (1986) 271–288.
• P. Donnelly, The heaps process, libraries and size biased permutations, J. Appl. Prob., 28, (1991) 322–335.
• P. Donnelly and P. Joyce, Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex, Stochastic Processes and their Applications, 31, (1989) 89–103.
• E.B. Dynkin, Some limit theorems for sums of independent random variables with infinite mathematical expectations, IMS-AMS Selected Translations in Math. Stat. and Probl, 1, (1961) 171–189.
• S. Engen, Stochastic Abundance Models with Emphasis on Biological Communities and Species Diversity, Chapman and Hall Ltd., (1978).
• W. J. Ewens, Population genetics theory - the past and the future, In S. Lessard, editor, Mathematical and Statistical Problems in Evolution, University of Montreal Press, Montreal, (1988).
• D. Freedman, On the asymptotic behavior of Bayes estimates in the discrete case, Ann. Math. Statist., 34, (1963) 1386–1403
• B. Fristedt, Intersections and limits of regenerative sets, In D. Aldous and R. Pemantle, editors, Random Discrete Structures, volume 76 of Mathematics and its Applications, Springer-Verlag, (1995).
• F.M. Hoppe, Size-biased filtering of Poisson-Dirichlet samples with an application to partition structures in genetics, Journal of Applied Probability, 23, (1986) 1008–1012.
• J. Horowitz, Semilinear Markov processes, subordinators and renewal theory, Z. Wahrsch. Verw. Gebiete, 24, (1972) 167–193.
• T. Ignatov, On a constant arising in the theory of symmetric groups and on Poisson-Dirichlet measures, Theory Probab. Appl., 27, (1982) 136–147.
• O. Kallenberg, The local time intensity of an exchangeable interval partition, In A. Gut and L. Holst, editors, Probability and Statistics, Essays in Honour of Carl-Gustav Essen, Uppsala University, (1983) 85–94.
• Olav Kallenberg, Splitting at backward times in regenerative sets, Annals of Probability, 9, (1981) 781–799.
• J.F.C. Kingman, Random discrete distributions, J. Roy. Statist. Soc. B, 37, (1975) 1–22.
• J.F.C. Kingman, The population structure associated with the Ewens sampling formula, Theor. Popul. Biol., 11, (1977) 274–283.
• J.F.C. Kingman, The representation of partition structures, J. London Math. Soc., 18, (1978) 374–380.
• F.B. Knight, Characterization of the Levy measure of inverse local times of gap diffusions, In Seminar on Stochastic Processes, 1981, Birkhauser, Boston, (1981) 53–78.
• S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, In Functional Analysis in Markov Processes, Lecture Notes in Math. 923, Springer, (1982) 235–249.
• J. Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc., 104, (1962) 62–78.
• J. Lamperti, Semi-stable Markov processes, I, Z. Wahrsch. Verw. Gebiete, 22, (1972) 205–225.
• B. Maisonneuve, Ensembles regeneratifs de la droite, Z. Wahrsch. Verw. Gebiete, 63, (1983) 501–510.
• J.W. McCloskey, A model for the distribution of individuals by species in an environment, Ph.D. thesis, Michigan State University, (1965).
• G.P. Patil and C. Taillie, Diversity as a concept and its implications for random communities, Bull. Int. Stat. Inst., 47, (1977) 497–515.
• M. Perman, Order statistics for jumps of normalized subordinators, Stoch. Proc. Appl., 46, (1993) 267–281.
• M. Perman, J. Pitman and M. Yor, Size-biased sampling of Poisson point processes and excursions, Probability Theory and Related Fields, 92, (1992) 21–39.
• J. Pitman, Stationary excursions, In Seminaire de Probabilites XXI, Lecture Notes in Math. 1247, Springer, (1986) 289–302.
• J. Pitman, Partition structures derived from Brownian motion and stable subordinators, Technical Report 346, Dept. Statistics, U.C. Berkeley, (1992). To appear in Bernoulli.
• J. Pitman, Random discrete distributions invariant under size-biased permutation, Technical Report 344, Dept. Statistics, U.C. Berkeley, (1992). To appear in Advances in Applied Probability.
• J. Pitman, The two-parameter generalization of Ewens' random partition structure, Technical Report 345, Dept. Statistics, U.C. Berkeley, (1992).
• J. Pitman and M. Yor, Arcsine laws and interval partitions derived from a stable subordinator, Proc. London Math. Soc.. (3), 65, (1992) 326–256.
• J. Pitman and M. Yor, On the relative lengths of excursions of some Markov processes, In preparation, (1995).
• J. Pitman and M. Yor, Some conditional expectation given an average of a stationary or self-similar random process, Technical Report 438, Dept. Statistics, U.C. Berkeley, (1995), In preparation.
• J. Pitman and M. Yor, The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, Technical Report 433, Dept. Statistics, U.C. Berkeley, (1995), To appear in The Annals of Probability.
• D. Revuz and M. Yor, Continuous martingales and Brownian motion, Springer, Berlin-Heidelberg, (1994), 2nd edition.
• M.S. Taqqu, A bibliographical guide to self-similar processes and long-range dependence, In Dependence in Probab. and Stat.: A Survey of Recent Results; Ernst Eberlein, Murad S. Taqqu (Ed.), Birkhauser (Basel, Boston), (1986), 137–162.
• A.M. Vershik, The asymptotic distribution of factorizations of natural numbers into prime divisors, Soviet Math. Dokl., 34, (1986) 57–61.
• A.M. Vershik and A.A. Shmidt, Limit measures arising in the theory of groups, I, Theor. Prob. Appl., 22, (1977) 79–85.
• A.M. Vershik and A.A. Shmidt, Limit measures arising in the theory of symmetric groups, II, Theor. Prob. Appl., 23, (1978) 36–49.
• J.G. Wendel, Zero-free intervals of semi-stable Markov processes, Math. Scand., 14, (1964) 21–34.