Distinguished properties of the gamma process and related topics

We study fundamental properties of the gamma process and their relation to various topics such as Poisson-Dirichlet measures and stable processes. We prove the quasi-invariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit of the stable processes and has an equivalent sigma-finite measure (quasi-Lebesgue) with important invariance properties. New properties of the gamma process can be applied to the Poisson-Dirichlet measures. We also emphasize the deep similarity between the gamma process and the Brownian motion. The connection of the above topics makes more transparent some old and new facts about stable and gamma processes, and the Poisson-Dirichlet measures.Comment: Prepublication du Laboratoire de Probabilites et Modeles Aleatoires, no. 575, Mars 200

Scaled subordinators and generalizations of the Indian buffet process

We study random families of subsets of $\mathbb{N}$ that are similar to exchangeable random partitions, but do not require constituent sets to be disjoint: Each element of ${\mathbb{N}}$ may be contained in multiple subsets. One class of such objects, known as Indian buffet processes, has become a popular tool in machine learning. Based on an equivalence between Indian buffet and scale-invariant Poisson processes, we identify a random scaling variable whose role is similar to that played in exchangeable partition models by the total mass of a random measure. Analogous to the construction of exchangeable partitions from normalized subordinators, random families of sets can be constructed from randomly scaled subordinators. Coupling to a heavy-tailed scaling variable induces a power law on the number of sets containing the first $n$ elements. Several examples, with properties desirable in applications, are derived explicitly. A relationship to exchangeable partitions is made precise as a correspondence between scaled subordinators and Poisson-Kingman measures, generalizing a result of Arratia, Barbour and Tavare on scale-invariant processes

On the Amount of Dependence in the Prime Factorization of a Uniform Random Integer

How much dependence is there in the prime factorization of a random integer distributed uniformly from 1 to n? How much dependence is there in the decomposition into cycles of a random permutation of n points? What is the relation between the Poisson-Dirichlet process and the scale invariant Poisson process? These three questions have essentially the same answers, with respect to total variation distance, considering only small components, and with respect to a Wasserstein distance, considering all components. The Wasserstein distance is the expected number of changes -- insertions and deletions -- needed to change the dependent system into an independent system. In particular we show that for primes, roughly speaking, 2+o(1) changes are necessary and sufficient to convert a uniformly distributed random integer from 1 to n into a random integer prod_{p leq n} p^{Z_p} in which the multiplicity Z_p of the factor p is geometrically distributed, with all Z_p independent. The changes are, with probability tending to 1, one deletion, together with a random number of insertions, having expectation 1+o(1). The crucial tool for showing that 2+epsilon suffices is a coupling of the infinite independent model of prime multiplicities, with the scale invariant Poisson process on (0,infty). A corollary of this construction is the first metric bound on the distance to the Poisson-Dirichlet in Billingsley's 1972 weak convergence result. Our bound takes the form: there are couplings in which E sum |log P_i(n) - (log n) V_i | = O(\log \log n), where P_i denotes the i-th largest prime factor and V_i denotes the i-th component of the Poisson-Dirichlet process. It is reasonable to conjecture that O(1) is achievable.Comment: 46 pages, appeared in Contemporary Combinatorics, 29-91, Bolyai Soc. Math. Stud., 10, Janos Bolyai Math. Soc., Budapest, 200

Gamma-Dirichlet Structure and Two Classes of Measure-valued Processes

The Gamma-Dirichlet structure corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the gamma process and vice versa. In this article, we begin with a brief review of existing results concerning the Gamma-Dirichlet structure. New results are obtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. The laws of the gamma process and the Dirichlet process are the respective reversible measures of the measure-valued branching diffusion with immigration and the Fleming-Viot process with parent independent mutation. We view the relation between these two classes of measure-valued processes as the dynamical Gamma-Dirichlet structure. Other results of this article include the derivation of the transition function of the Fleming-Viot process with parent independent mutation from the transition function of the measure-valued branching diffusion with immigration, and the establishment of the reversibility of the latter. One of these is related to an open problem by Ethier and Griffiths and the other leads to an alternative proof of the reversibility of the Fleming-Viot process.Comment: 23 page

Extensions of Billingsley's Theorem via Multi-Intensities

Let $p_1 \ge p_2 \ge \dots$ be the prime factors of a random integer chosen uniformly from $1$ to $n$, and let $$\frac{\log p_1}{\log n}, \frac{\log p_2}{\log n}, \dots$$ be the sequence of scaled log factors. Billingsley's Theorem (1972), in its modern formulation, asserts that the limiting process, as $n \to \infty$, is the Poisson-Dirichlet process with parameter $\theta =1$. In this paper we give a new proof, inspired by the 1993 proof by Donnelly and Grimmett, and extend the result to factorizations of elements of normed arithmetic semigroups satisfying certain growth conditions, for which the limiting Poisson-Dirichlet process need not have $\theta =1$. We also establish Poisson-Dirichlet limits, with $\theta \ne 1$, for ordinary integers conditional on the number of prime factors deviating from the usual value $\log \log n$. At the core of our argument is a purely probabilistic lemma giving a new criterion for convergence in distribution to a Poisson-Dirichlet process, from which the number-theoretic applications follow as straightforward corollaries. The lemma uses ingredients similar to those employed by Donnelly and Grimmett, but reorganized so as to allow subsequent number theory input to be processed as rapidly as possible. A by-product of this work is a new characterization of Poisson-Dirichlet processes in terms of multi-intensities.Comment: 25 page

Universal statistics of vortex tangle in three-dimensional wave chaos

The tangled nodal lines (wave vortices) in random, three-dimensional wavefields are studied as an exemplar of a fractal loop soup. Their statistics are a three-dimensional counterpart to the characteristic random behaviour of nodal domains in quantum chaos, but in three-dimensions the filaments can wind around one another to give distinctly different large scale behaviours. By tracing numerically the structure of the vortices, their conformations are shown to follow recent analytical predictions for random vortex tangles with periodic boundaries, where the local disorder of the model `averages out' to produce large scale power law scaling relations whose universality classes do not depend on the local physics. These results explain previous numerical measurements in terms of an explicit effect of the periodic boundaries, where the statistics of the vortices are strongly affected by the large scale connectedness of the system even at arbitrarily high energies. The statistics are investigated primarily for static (monochromatic) wavefields, but the analytical results are further shown to directly describe the reconnection statistics of vortices evolving in certain dynamic systems, or occurring during random perturbations of the static configuration.Comment: 18 pages, 5 figure

A Simple Direct Proof of Billingsley's Theorem

Billingsley's theorem (1972) asserts that the Poisson--Dirichlet process is the limit, as $n \to \infty$, of the process giving the relative log sizes of the largest prime factor, the second largest, and so on, of a random integer chosen uniformly from 1 to $n$. In this paper we give a new proof that directly exploits Dickman's asymptotic formula for the number of such integers with no prime factor larger than $n^{1/u}$, namely $\Psi(n,n^{1/u}) \sim n \rho(u)$, to derive the limiting joint density functions of the finite-dimensional projections of the log prime factor processes. Our main technical tool is a new criterion for the convergence in distribution of non-lattice discrete random variables to continuous random variables.Comment: 13 page

Poincare Inequality on the Path Space of Poisson Point Processes

The quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O-U Dirichlet form on the Wiener space satisfying the log-Sobolev inequality, this Dirichlet form merely satisfies the Poincare inequality but not the log-Sobolev one

Negative Binomial Process Count and Mixture Modeling

The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an NB process for count modeling. A draw from the NB process consists of a Poisson distributed finite number of distinct atoms, each of which is associated with a logarithmic distributed number of data samples. We reveal relationships between various count- and mixture-modeling distributions and construct a Poisson-logarithmic bivariate distribution that connects the NB and Chinese restaurant table distributions. Fundamental properties of the models are developed, and we derive efficient Bayesian inference. It is shown that with augmentation and normalization, the NB process and gamma-NB process can be reduced to the Dirichlet process and hierarchical Dirichlet process, respectively. These relationships highlight theoretical, structural and computational advantages of the NB process. A variety of NB processes, including the beta-geometric, beta-NB, marked-beta-NB, marked-gamma-NB and zero-inflated-NB processes, with distinct sharing mechanisms, are also constructed. These models are applied to topic modeling, with connections made to existing algorithms under Poisson factor analysis. Example results show the importance of inferring both the NB dispersion and probability parameters.Comment: To appear in IEEE Trans. Pattern Analysis and Machine Intelligence: Special Issue on Bayesian Nonparametrics. 14 pages, 4 figure

A Class of Infinite Dimensional Diffusion Processes with Connection to Population Genetics

Starting from a sequence of independent Wright-Fisher diffusion processes on $[0,1]$, we construct a class of reversible infinite dimensional diffusion processes on \DD_\infty:= \{{\bf x}\in [0,1]^\N: \sum_{i\ge 1} x_i=1\} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence to the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space $S$. This provides a reasonable alternative to the Fleming-Viot process which does not satisfy the log-Sobolev inequality when $S$ is infinite as observed by W. Stannat \cite{S}.Comment: 14 page