arXiv subjects

Jordan M. Stoyanov

Publications and source records attributed to Jordan M. Stoyanov.

6 recordsLinked to original sources

arXiv2025

New Sufficient Conditions for Moment-determinacy via Probability Density Tails

One of the ways to characterize a probability distribution is to show that it is moment-determinate, uniquely determined by knowing all its moments. The uniqueness, in the absolutely continuous case, depends entirely on the behaviour of the tails of the density function f. We find and exploit a condition, (D), in terms only of f which is of a `general' form and easy to check. Condition (D), showing the `speed' for f to tend to zero, is sufficient to conclude the moment determinacy. We establish a series of theorems and corollaries in both Stieltjes and Hamburger cases and provide an interesting illustrative example. The results in this paper are either new or extend some recently published results.

math.PR↗

arXiv2023

The Problem of Moments: A Bunch of Classical Results With Some Novelties

We summarize significant classical results on (in)determinacy of measures in terms of their finite positive integer order moments. Well-known is the role of the smallest eigenvalues of Hankel matrices, starting from Hamburger's results a century ago and ending with the great progress made only in recent times by C. Berg and collaborators. We describe here known results containing necessary and sufficient conditions for moment (in)determinacy in both Hamburger and Stieltjes moment problems. In our exposition we follow an approach different from that commonly used. There are novelties well complementing the existing theory. Among them are: (a) to emphasize on the geometric interpretation of the indeterminacy conditions; (b) exploit fine properties of the eigenvalues of perturbed symmetric matrices allowing to derive new lower bounds for the smallest eigenvalues of Hankel matrices; these bounds are used for concluding indeterminacy; (c) provide new arguments to confirm classical results; (d) give new numerical illustrations involving commonly used probability distributions.

math.FA↗

arXiv2022

New Characterizations of the Gamma Distribution via Independence of Two Statistics by Using Anosov's Theorem

Available in the literature are properties which characterize the gamma distribution via independence of two appropriately chosen statistics. Well-known is the classical result when one of the statistics is the sample mean and the other one the sample coefficient of variation. In this paper, we elaborate on a version of Anosov's theorem which allows to establish a general result, Theorem 1, and a series of seven corollaries providing new characterization results for gamma distributions. We keep the sample mean as one of involved statistics, while now the second one can be taken from a quite large class of homogeneous feasible definite statistics. It is relevant to mention that there is an interesting parallel between the new characterization results for gamma distributions and recent characterization results for the normal distribution.

math.PR↗

arXiv2020

Characterization of Probability Distributions via Functional Equations of Power-Mixture Type

We study power-mixture type functional equations in terms of Laplace-Stieltjes transforms of probability distributions. These equations arise when studying distributional equations of the type Z = X + TZ, where T is a known random variable, while the variable Z is defined via X, and we want to `find' X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results which are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics.

math.PR↗

arXiv2020

New Checkable Conditions for Moment Determinacy of Probability Distributions

We have analyzed some conditions which are essentially involved in deciding whether or not a probability distribution is unique (moment-determinate) or non-unique (moment-indeterminate) by its moments. We suggest new conditions concerning both absolutely continuous and discrete distributions. By using the new conditions, which are easily checkable, we either establish new results, or extend previous ones in both Hamburger case (distributions on the whole real line) and Stieltjes case (distributions on the positive half-line). Specific examples illustrate both the results and the relationship between the new conditions and previously available conditions.

math.PR↗

arXiv2019

On conditions under which a probability distribution is uniquely determined by its moments

We study the relationship between the well-known Carleman's condition guaranteeing that a probability distribution is uniquely determined by its moments, and a recent easily checkable condition on the rate of growth of the moments. We use asymptotic methods in theory of integrals and involve properties of the Lambert $W$-function to show that the quadratic rate of growth of the ratios of consecutive moments, as a sufficient condition for uniqueness, is more restrictive than Carleman's condition. We derive a series of statements, one of them showing that Carleman's condition does not imply Hardy's condition, although the inverse implication is true. Related topics are also discussed.

math.PR↗