arXiv subjects

Hristo Sariev

Publications and source records attributed to Hristo Sariev.

6 recordsLinked to original sources

arXiv2026

Some developments of exchangeable measure-valued Pólya sequences

Measure-valued Pólya sequences (MVPS) are processes whose dynamics are governed by generalized Pólya urn schemes with infinitely many colors. Assuming a general reinforcement rule, exchangeable MVPSs can be viewed as extensions of Blackwell and MacQueen's Pólya sequence, which characterizes an exchangeable sequence whose directing random measure has a Dirichlet process prior distribution. Here, we show that the prior distribution of any exchangeable MVPS is a Dirichlet process mixture with respect to a latent parameter that is associated with the atoms of an emergent conditioning $σ$-algebra. As the mixing components have disjoint supports, the directing random measure can be interpreted as a random histogram with bins randomly located on these same atoms. Furthermore, we extend the basic exchangeable MVPS to include a null component in the reinforcement, which corresponds to the presence of a fixed component in the directing random measure. Finally, we examine the effects of relaxing exchangeability to conditional identity in distribution (c.i.d.) and find out that the two are equivalent for balanced MVPSs. The paper features a complementary study of some properties of probability kernels that underlies the analysis of exchangeable and c.i.d. MVPSs.

math.PR↗

arXiv2025

On some properties of probability kernels

Although regular conditional distributions (r.c.d.) are well-defined and widely used measure-theoretic objects, they can violate our intuition from the classical definition of a conditional probability given an event. For that purpose, the notion of a proper r.c.d. has been introduced. Here, we study how properness, viewed as a property of probability kernels in general, is related to stationarity, compatibility, reversibility and totality, revealing the effects these properties have on the structure of probability kernels. As a further development, we consider the inverse problem of characterizing certain classes of r.c.d.s in terms of the above properties. In particular, we derive necessary and sufficient conditions under which, for a given probability kernel, there exists a unique (in some sense) sub-$σ$-algebra such that the probability kernel is a proper r.c.d. given that sub-$σ$-algebra.

math.PR↗

arXiv2024

Sufficientness postulates for measure-valued Pólya urn sequences

In a recent paper, the authors studied the distribution properties of a class of exchangeable processes, called measure-valued Pólya sequences (MVPS), which arise as the observation process in a generalized urn sampling scheme. Here we present several results in the form of "sufficientness" postulates that characterize their predictive distributions. In particular, we show that exchangeable MVPSs are the unique exchangeable models whose predictive distributions are a mixture of the marginal distribution and the average of a probability kernel evaluated at past observation. When the latter coincides with the empirical measure, we recover a well-known result for the exchangeable model with a Dirichlet process prior. In addition, we provide a "pure" sufficientness postulate for exchangeable MVPSs that does not assume a particular analytic form for the predictive distributions. Two other sufficientness postulates consider the case when the state space is finite.

math.PR↗

arXiv2024

Characterization of exchangeable measure-valued Pólya urn sequences

Measure-valued Pólya urn sequences (MVPS) are a generalization of the observation processes generated by $k$-color Pólya urn models, where the space of colors $\mathbb{X}$ is a complete separable metric space and the urn composition is a finite measure on $\mathbb{X}$, in which case reinforcement reduces to a summation of measures. In this paper, we prove a representation theorem for the reinforcement measures $R$ of all exchangeable MVPSs, which leads to a characterization result for their directing random measures $\tilde{P}$. In particular, when $\mathbb{X}$ is countable or $R$ is dominated by the initial distribution $ν$, then any exchangeable MVPS is a Dirichlet process mixture model over a family of probability distributions with disjoint supports. Furthermore, for all exchangeable MVPSs, the predictive distributions converge on a set of probability one in total variation to $\tilde{P}$. Importantly, we do not restrict our analysis to balanced MVPSs, in the terminology of $k$-color urns, but rather show that the only non-balanced exchangeable MVPSs are sequences of i.i.d. random variables.

math.PR↗

arXiv2023

Infinite-color randomly reinforced urns with dominant colors

We define and prove limit results for a class of dominant Pólya sequences, which are randomly reinforced urn processes with color-specific random weights and unbounded number of possible colors. Under fairly mild assumptions on the expected reinforcement, we show that the predictive and the empirical distributions converge almost surely (a.s.) in total variation to the same random probability measure $\tilde{P}$; moreover, $\tilde{P}(\mathcal{D})=1$ a.s., where $\mathcal{D}$ denotes the set of dominant colors for which the expected reinforcement is maximum. In the general case, the predictive probabilities and the empirical frequencies of any $δ$-neighborhood of $\mathcal{D}$ converge a.s. to one. That is, although non-dominant colors continue to be regularly observed, their distance to $\mathcal{D}$ converges in probability to zero. We refine the above results with rates of convergence. We further hint potential applications of dominant Pólya sequences in randomized clinical trials and species sampling, and use our central limit results for Bayesian inference.

math.PR↗

arXiv2023

Moments of exponential functionals of Lévy processes on a deterministic horizon -- identities and explicit expressions

In this work, we consider moments of exponential functionals of Lévy processes on a deterministic horizon. We derive two convolutional identities regarding these moments. The first one relates the complex moments of the exponential functional of a general Lévy process up to a deterministic time to those of the dual Lévy process. The second convolutional identity links the complex moments of the exponential functional of a Lévy process, which is not a compound Poisson process, to those of the exponential functionals of its ascending/descending ladder heights on a random horizon determined by the respective local times. As a consequence, we derive a universal expression for the half-negative moment of the exponential functional of any symmetric Lévy process, which resembles in its universality the passage time of symmetric random walks. The $(n-1/2)^{th}$, $n\geq 0$ moments are also discussed. On the other hand, under extremely mild conditions, we obtain a series expansion for the complex moments (including those with negative real part) of the exponential functionals of subordinators. This significantly extends previous results and offers neat expressions for the negative real moments. In a special case, it turns out that the Riemann zeta function is the minus first moment of the exponential functional of the Gamma subordinator indexed in time.

math.PR↗