Linear functionals and Markov chains associated with Dirichlet processes (original) (raw)

functionals of Dirichlet processes

2000

The paper deals with the approximation of the law of a random functional of a Dirich- let process using a flnite number of its moments. In particular, three classes of approx- imation procedures { expansions in series of orthonormal polynomials, the maximum entropy method and mixtures of known distributions { are discussed. A comparison of the difierent approximation procedures is

On the Convergence of Dirichlet Processes

Bernoulli, 1999

For a given weakly convergent sequence fX n g of Dirichlet processes we show weak convergence of the sequence of the corresponding quadratic variation processes as well as stochastic integrals driven by the X n values provided that the condition UTD (a counterpart to the condition UT for Dirichlet processes) holds true. Moreover, we show that under UTD the limit process of fX n g is a Dirichlet process, too.

On Non-Continuous Dirichlet Processes

2003

We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C 1 transformation.

Application to windows of Dirichlet processes

2011

This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of Métivier and Pellaumail which is quite restrictive. We make use of the notion of χ-covariation which is a generalized notion of covariation for processes with values in two Banach spaces B1 and B2. χ refers to a suitable subspace of the dual of the projective tensor product of B1 and B2. We investigate some C 1 type transformations for various classes of stochastic processes admitting a χ-quadratic variation and related properties. If X 1 and X 2 admit a χ-covariation, F i : Bi → R, i = 1, 2 are of class C 1 with some supplementary assumptions then the covariation of the real processes F 1 (X 1) and F 2 (X 2) exist. A detailed analysis will be devoted to the so-called window processes. Let X be a real continuous process; the C([−τ, 0])-valued process X(•) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0], is called window process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. In fact we aim to generalize the following properties valid for B = R. If X = X is a real valued Dirichlet process and F : B → R of class C 1 (B) then F (X) is still a Dirichlet process. If X = X is a weak Dirichlet process with finite quadratic variation, and F : C 0,1 ([0, T ] × B) is of class C 0,1 , then (F (t, Xt)) is a weak Dirichlet process. We specify corresponding results when B = C([−τ, 0]) and X = X(•). This will consitute a significant Fukushima decomposition for functionals of windows of (weak) Dirichlet processes. As applications, we give a new technique for representing path-dependent random variables.

Infinite dimensional weak Dirichlet processes and convolution type processes

Stochastic Processes and their Applications, 2017

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Banach space H, is the sum of a local martingale and a suitable orthogonal process. The concept of weak Dirichlet process fits the notion of convolution type processes, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs). In particular the mentioned decomposition appears to be a substitute of an Itô's type formula applied to f (t, X(t)) where f : [0, T ] × H → R is a C 0,1 function and X a convolution type processes.

Theory and numerical analysis for exact distributions of functionals of a Dirichlet process

The Annals of Statistics, 2002

The distribution of a mean or, more generally, of a vector of means of a Dirichlet process is considered. Some characterizing aspects of this paper are: (i) a review of a few basic results, providing new formulations free from many of the extra assumptions considered to date in the literature, and giving essentially new, simpler and more direct proofs; (ii) new numerical evaluations, with any prescribed error of approximation, of the distribution at issue; (iii) a new form for the law of a vector of means. Moreover, as applications of these results, we give: (iv) the sharpest condition sufficient for the distribution of a mean to be symmetric; (v) forms for the probability distribution of the variance of the Dirichlet random measure; (vi) some hints for determining the finite-dimensional distributions of a random function connected with Bayesian methods for queuing models.

On Dependent Dirichlet Processes for General Polish Spaces

arXiv (Cornell University), 2022

We study Dirichlet process-based models for sets of predictor-dependent probability distributions, where the domain and predictor space are general Polish spaces. We generalize the definition of dependent Dirichlet processes, originally constructed on Euclidean spaces, to more general Polish spaces. We provide sufficient conditions under which dependent Dirichlet processes have appealing properties regarding continuity (weak and strong), association structure, and support (under different topologies). We also provide sufficient conditions under which mixture models induced by dependent Dirichlet processes have appealing properties regarding strong continuity, association structure, support, and weak consistency under i.i.d. sampling of both responses and predictors. The results can be easily extended to more general dependent stick-breaking processes.

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 proce...

Special weak Dirichlet processes and BSDEs driven by a random measure

Bernoulli, 2018

This paper considers a forward BSDE driven by a random measure, when the underlying forward process X is a special semimartingale, or even more generally, a special weak Dirichlet process. Given a solution (Y, Z, U), generally Y appears to be of the type u(t, X t) where u is a deterministic function. In this paper we identify Z and U in terms of u applying stochastic calculus with respect to weak Dirichlet processes.

On L1-weak ergodicity of markov processes

2013

In the present paper we investigate the L 1-weak ergodicity of nonhomogeneous discrete Markov processes with general state spaces. Note that the L 1-weak ergodicity is weaker than well-known weak ergodicity. It was known a necessary and sufficient condition for such processes to satisfy the weak ergodicity. In this paper we provide a necessary and sufficient condition for nonhomogeneous discrete Markov processes to satisfy the L 1-weak ergodicity. Moreover, as an application of the main result, we provide more concrete examples of Markov processes which satisfy the L 1-weak ergodicity