On a characterization of convolutions of Gaussian and Haar distributions

We prove some analogues of the well‐known Skitovich–Darmois and Heyde characterization theorems for a second countable locally compact Abelian group X under the assumption that the distributions of the random variables have continuous positive densities with respect to a Haar measure on X and the coefficients in the linear forms considered are topological automorphisms of X.     

On a characterization of rectangular distributions

Let (X₍₁₎, X₍₂₎) be the order statistics of a sample of size 2 from a population having density ?. It is well known that X₍₁₎ and X₍₂₎ are positively correlated. We show that cov(X₍₁₎, X₍₂₎) has an upper bound which is attained if and only if ? is rectangular density on (0, 1). Our proof uses a 2-dimensional extension of a result due to Polya.     

On the geometry of generalized Gaussian distributions

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q=1 case these are the normal distributions. Some methods to endow this parameter space with a Riemannian metric is presented: the second derivative of the q-Rényi entropy, the Tsallis entropy, and the relative entropy give rise to a Riemannian metric, the Fisher information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovri?, Min-Oo and Ruh. These metrics are different; therefore, our differential geometrical calculations are based on a new metric with parameters, which covers all the above-mentioned metrics for special values of the parameters, among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors, and the scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metrics in quantum information geometry can be easily recovered from classical metrics.     

The convolution of functions and distributions

The non-commutative convolution f∗g of two distributions f and g in D^′ is defined to be the limit of the sequence {(fτn)∗g}, provided the limit exists, where {τn} is a certain sequence of functions in D converging to 1. It is proved that

     

On the titchmarsh convolution theorem for distributions on the circle

We prove a version of the Titchmarsh convolution theorem for distributions on the circle. We show that a certain “naïve” form of the Titchmarsh theorem can be violated, but only for the convolution of distributions with certain symmetry properties.     

Decompositions of stochastic convolution driven by a white-fractional Gaussian noise

Let u=\{u(t,x);(t,x)\in {ℝ}_{+}\times ℝ\}be the solution to a linear stochastic heat equation driven by a Gaussian noise, which is a Brownian motion in time and a fractional Brownian motion in space with Hurst parameterH\in (\mathrm{0},\mathrm{1}): For any givenx\in ℝ(\text{resp}\text{.},t\in {ℝ}_{+}), we show a decomposition of the stochastic processt\to u(t,x)(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.,x\to u(t,x))as the sum of a fractional Brownian motion with Hurst parameter H/2 (resp., H) and a stochastic process with C^∞-continuous trajectories. Some applications of those decompositions are discussed.     

On limit distributions of first crossing points of Gaussian sequences

Let {X_k, k?Z} be a stationary Gaussian sequence with EX₁ – 0, EX²_k = 1 and EX₀X_k = r_k. Define τ_x = inf{k: X_k >– βk} the first crossing point of the Gaussian sequence with the function – βt (β > 0). We consider limit distributions of τ_x as β→0, depending on the correlation function r_k. We generalize the results for crossing points τ_x = inf{k: X_k >β?(k)} with ?(– t)?t^γL(t) for t→∞, where γ > 0 and L(t) varies slowly.     

Statistical estimation of a mixture of Gaussian distributions

The paper is devoted to the problem of statistical estimation of a multivariate distribution density, which is a discrete mixture of Gaussian distributions. A heuristic approach is considered, based on the use of the EM algorithm and nonparametric density estimation with a sequential increase in the number of components of the mixture. Criteria for testing of model adequacy are discussed.     

On the Wick theorem for mixtures of centered Gaussian distributions

We consider centered conditionally Gaussian d-dimensional vectors X with random covariance matrix Ξ having an arbitrary probability distribution law on the set of nonnegative definite symmetric d × d matrices M _d ⁺. The paper deals with the evaluation problem of mean values \( E\left[ {\prod\nolimits_{i = 1}^{2n} {\left( {{c_i},X} \right)} } \right] \) for c _i ∈ ? ^d , i = 1, …, 2n, extending the Wick theorem for a wide class of non-Gaussian distributions. We discuss in more detail the cases where the probability law ?(Ξ) is infinitely divisible, the Wishart distribution, or the inverse Wishart distribution. An example with Ξ \( = \sum\nolimits_{j = 1}^m {{Z_j}{\sum_j}} \), where random variables Z _j , j = 1, …, m, are nonnegative, and Σ _j ∈ M _d ⁺, j = 1, …, m, are fixed, includes recent results from Vignat and Bhatnagar, 2008.     

Changing the variable in convolution of distributions

In this paper the author has extended the concept of changing the variables in distributions to the convolution of distributions. For an infinitely differentiable functionh(x), he has first defined the convolution of two distributions f(h(x)) and g(h(x)) and then proved some of its properties. Finally, he has applied his results to the fractional integral and fractional differential operators     

On a variety of commutative multiplicatively idempotent semirings

We prove that the variety \({{\mathscr {V}}}\) of commutative multiplicatively idempotent semirings satisfying \(x+y+xyz\approx x+y\) is generated by a single three-element semiring. Moreover, we describe a normal form system for terms in \({{\mathscr {V}}}\) and we show that the word problem in \({{\mathscr {V}}}\) is solvable. Although \({{\mathscr {V}}}\) is locally finite, it is residually big.     

On rapid idempotent ultrafilters

This short note contains the proofs of two small but somewhat surprising results about ultrafilters on \(\mathbb {N}\) : (1) strongly summable ultrafilters are rapid, (2) every rapid ultrafilter induces a closed left ideal of rapid ultrafilters. As a consequence, there will be rapid minimal idempotents in all models of set theory with rapid ultrafilters. The history of this result has been published as an experiment in mathematical writing on the author’s website (Krautzberger, One Day in Colorado or Strongly Summable Ultrafilters are Rapid, 2012) and (Krautzberger, Rapid Idempotent Ultrafilters, 2012) where you can can also find additional remarks by Blass and Hindman, offering a form of peer-review.     

Hankel convolution operators on entire functions and distributions

In this paper we study the Hankel convolution operators on the space of even and entire functions and on Schwartz distribution spaces. We characterize the Hankel convolution operators as those ones that commute with Hankel translations and with a Bessel operator. Also we prove that the Hankel convolution operators are hypercyclic and chaotic on the spaces under consideration.     

Multi-variable subordination distributions for free additive convolution

Let k be a positive integer and let Dc(k) denote the space of joint distributions for k-tuples of selfadjoint elements in C^∗-probability space. The paper studies the concept of “subordination distribution of μ?ν with respect to ν” for μ,ν∈Dc(k), where ? is the operation of free additive convolution on Dc(k). The main tools used in this study are combinatorial properties of R-transforms for joint distributions and a related operator model, with operators acting on the full Fock space.Multi-variable subordination turns out to have nice relations to a process of evolution towards ?-infinite divisibility on Dc(k) that was recently found by Belinschi and Nica (arXiv: 0711.3787). Most notably, one gets better insight into a connection which this process was known to have with free Brownian motion.