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.
The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixtures of probability distributions, also known as mixed distributions, are an exciting new area for optimal quantization. In this paper, we investigate the optimal quantization for three different mixed distributions generated by uniform distributions associated with probability vectors.
We show existence of an invariant probability measure for a class of functional McKean-Vlasov SDEs by applying Kakutani's fixed point theorem to a suitable class of probability measures on a space of continuous functions. Unlike some previous works, we do not assume a monotonicity condition to hold. Further, our conditions are even weaker than some results in the literature on invariant probability measures for functional SDEs without dependence on the law of the solution.
Let $\bf{x}$ be a random variable with density $ρ(x)$ taking values in ${\mathbb R}^d$. We are interested in finding a representation for the shape of $ρ(x)$, i.e. for the orbit $\{ ρ(g\cdot x) | g\in E(d) \}$ of $ρ$ under the Euclidean group. Let $x_1$ and $x_2$ be two random samples picked, independently, following $ρ(x)$, and let $Δ$ be the squared Euclidean distance between $x_1$ and $x_2$. We show, if $ρ(x)$ is a mixture of Gaussians whose covariance matrix is the identity, and if the means of the Gaussians are in generic position, then the density $ρ(x)$ is reconstructible, up to a rigid…
Given a purely atomic probability measure with support on n points, P, any mean-preserving contraction (mpc) of P, Q, with support on m > n points is a mixture of mpcs of P, each with support on most n points. We illustrate an application of this result in economics.
This paper investigates probability density functions (PDFs) that are continuous everywhere, nearly uniform around the mode of distribution, and adaptable to a variety of distribution shapes ranging from bell-shaped to rectangular. From the viewpoint of computational tractability, the PDF based on the Fermi-Dirac or logistic function is advantageous in estimating its shape parameters. The most appropriate PDF for $n$-variate distribution is of the form: $p\left(\mathbf{x}\right)\propto\left[\cosh\left(\left[\left(\mathbf{x}-\mathbf{m}\right)^{\mathsf{T}}\boldsymbolΣ^{-1}\left(\mathbf{x}-\mathb…
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室有限公司）；Tokyo（东京）
The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed-form formula for Gaussian mixture models when parameterized by integer exponents. This result extends to arbitrary mixtures of exponential families with natural parameter spaces being cones: This includes the binomial, the multinomial, the zero-centered Laplacian, the Gaussian a…
In this paper we introduce the theory of derivatives of moments and (moment) functionals to represent moment functionals by Gaussian mixtures, characteristic functions of polytopes, and simple functions of polytopes. We study, among other measures, Gaussian mixtures, their reconstruction from moments and especially the number of Gaussians needed to represent moment functionals. We find that there are moment functionals $L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\to\mathbb{R}$ which can be represented by a sum of $\binom{n+2d}{n} - n\cdot \binom{n+d}{n} + \binom{n}{2}$ Gaussians but not less. Hence,…
We offer two novel characterizations of the Zeta distribution: first, as tractable continuous mixtures of Negative Binomial distributions (with fixed shape parameter, r > 0), and second, as a tractable continuous mixture of Poisson distributions. In both the Negative Binomial case for r >= 1 and the Poisson case, the resulting Zeta distributions are identifiable because each mixture can be associated with a unique mixing distribution. In the Negative Binomial case for 0 < r < 1, the mixing distributions are quasi-distributions (for which the quasi-probability density function assumes some nega…
