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Kazuki Okamura

Publications and source records attributed to Kazuki Okamura.

At least 19 recordsLinked to original sources
arXiv2026

The two-dimensional Matkowski--Sut\^o equation with holomorphic and strictly increasing generators

We study the two-dimensional Matkowski--Sut\^o equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of T\'oth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Dar\'oczy and P\'ales, persists under holomorphy but not under monotonicity.

math.CV↗
arXiv2026

Dual-connection midpoint matrix means and their Gauss composition

Nakamura [J. Comput. Appl. Math. 131 (2001)] proved that the arithmetic--harmonic matrix iteration converges quadratically to the geometric matrix mean, the Riemannian midpoint of the affine-invariant metric on positive-definite matrices. We ask, more generally, when a Riemannian midpoint is a Gauss compound mean. A Riemannian metric and an affine connection induce three midpoint maps, for the connection, its metric dual, and the Levi--Civita connection. The Gauss composition of the first two equals the Levi--Civita midpoint exactly when the latter is invariant under one step of the iteration, and this invariance holds whenever an isometry acts as the point reflection about the Levi--Civita midpoint and exchanges the connection with its dual; for sufficiently close initial pairs it implies quadratic convergence of the iterations. Nakamura's iteration is the model case, and the criterion extends it to a one-parameter family of matrix iterations whose Gauss composition is again the geometric matrix mean. A dually flat Hessian metric shows that duality alone does not suffice, while Euclidean metrics with a parallel cubic form provide non-flat examples in every dimension larger than one. In dimension one, the invariance is characterized completely through the Matkowski--Suto equation, and we conjecture that for Euclidean dual pairs it forces the cubic form to be constant, which we prove in dimension one and for scalar multiples of a constant cubic form.

math.DG↗
arXiv2026

The Moran--Hutchinson formula in semimetric spaces

We establish the Moran--Hutchinson formula for attractors of finite systems of surjective similitudes on semimetric spaces. More precisely, for a complete, normal semimetric space satisfying strong regularity and geometric doubling, we prove that the open set condition implies that the Hausdorff measure of the attractor at the similarity dimension is positive and finite. We also prove the converse, specifically, positivity of the Hausdorff measure at the similarity dimension implies the open set condition. This provides a partial answer to a question posed by Bessenyei and P\'{e}nzes in 2022.

math.DS↗
arXiv2026

Group invariance of $f$-divergences and the Fisher--Rao distance

Many statistical models have natural symmetries described by a group action. We study how such symmetries affect the comparison of two distributions. We work with a transformation model in which a group acts on both the sample space and the parameter space, and the densities transform with a multiplier. Under this assumption, we show that every $f$-divergence is invariant under the group action. As a consequence, an invariant divergence depends only on a maximal invariant of the pair of parameters. When the action on the parameter space is transitive, this maximal invariant is given by a double coset. We apply this result to multidimensional location-scale families, and we show that the same reduction applies to the Fisher--Rao distance.

math.ST↗
arXiv2026

Asymptotic estimates for multiple point ranges of transient random walks on graphs

We study multiple point ranges for random walks on graphs, extending known asymptotic results obtained for random walks on groups. A distinctive feature is that algebraic and translation-invariance assumptions are replaced by a uniform tail condition on the first return time to the starting point. Under this condition, we obtain upper and lower bounds of linear order for the expectations of the number of sites visited at least a given number of times and the number of sites visited exactly that number of times. We also prove the corresponding almost sure bounds under a stronger condition. In spatially homogeneous transient cases, these bounds coincide and yield a strong law of large numbers. We apply these estimates to derive asymptotic results for functions of the local times.

math.PR↗
arXiv2026

Singularity for graph-directed conjugate equations indexed by a two-vertex digraph

We study graph-directed conjugate functional equations on the unit interval indexed by the complete digraph with self-loops on two vertices. We focus on the singularity and regularity of the solutions for compatible systems of weak contractions. First, we show that both solutions are singular in the affine case unless the two systems coincide; second, we obtain a dichotomy between singularity and smoothness for a class of linear fractional systems; and finally, we give a sufficient condition for singularity in a non-linear setting.

math.DS↗
arXiv2025

Asymptotics of the maximum likelihood estimator of the location parameter of Pearson Type VII distribution

We study the maximum likelihood estimator of the location parameter of the Pearson Type VII distribution with known scale. We rigorously establish precise asymptotic properties such as strong consistency, asymptotic normality, Bahadur efficiency and asymptotic variance of the maximum likelihood estimator. Our focus is the heavy-tailed case, including the Cauchy distribution. The main difficulty lies in the fact that the likelihood equation may have multiple roots; nevertheless, the maximum likelihood estimator performs well for large samples.

math.ST↗
arXiv2025

Construction of graph-directed invariant sets of weak contractions on semi-metric spaces

We present a construction of graph-directed invariant sets of weak contractions in the sense of Matkowski-Rus on semi-metric spaces. We follow the approach by Bessenyei and P\'enzes, which applies the Kuratowski noncompactness measure without relying on Blascke's completeness theorem. We also establish a relationship between this approach and a generalized de Rham's functional equation indexed by a finite directed graph.

math.MG↗
arXiv2024

Quantitative estimates for singularity for conjugate equations driven by linear fractional transformations

We consider the conjugate equation driven by two families of finite maps on the unit interval satisfying a compatibility condition. This framework contains de Rham's functional equations. We give sufficient conditions for singularity of the solution with quantitative estimates in the case where the equation is driven by a family of non-affine maps and a family of linear fractional transformations.

math.CA↗
arXiv2023

Power means of random variables and characterizations of distributions via fractional calculus

We investigate fractional moments and expectations of power means of complex-valued random variables by using fractional calculus. We deal with both negative and positive orders of the fractional derivatives. The one-dimensional distributions are characterized in terms of the fractional moments without any moment assumptions. We explicitly compute the expectations of the power means for both the univariate Cauchy distribution and the Poincar\'e distribution on the upper-half plane. We show that for these distributions the expectations are invariant with respect to the sample size and the value of the power.

math.PR↗
arXiv2023

Properties of complex-valued power means of random variables and their applications

We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is an example of a homogeneous quasi-arithmetic mean. Under certain conditions, several limit theorems hold for the power mean, similar to the case of the arithmetic mean of i.i.d. integrable random variables. Our feature is that the generators of the power means are allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. We establish integrabilities of the power mean of i.i.d. non-integrable random variables and a limit theorem for the variances of the power mean. We also consider the behavior of the power mean as the parameter of the power varies. The complex-valued power means are unbiased, strongly-consistent, robust estimators for the joint of the location and scale parameters of the Cauchy distribution.

math.PR↗
arXiv2023

Metrization of powers of the Jensen-Shannon divergence

Metrization of statistical divergences is valuable in both theoretical and practical aspects. One approach to obtaining metrics associated with divergences is to consider their fractional powers. Motivated by this idea, Os\'an, Bussandri, and Lamberti (2018) studied the metrization of fractional powers of the Jensen-Shannon divergence between multinomial distributions and posed an open problem. In this short note, we provide an affirmative answer to their conjecture. Moreover, our method is also applicable to fractional powers of $f$-divergences between Cauchy distributions.

cs.IT↗
arXiv2022

Confidence disc and square for Cauchy distributions

We will construct a confidence region of parameters for a sample of size $N$ from Cauchy distributed random variables. Although Cauchy distribution has two parameters, a location parameter $μ\in \mathbb{R}$ and a scale parameter $σ> 0$, we will infer them at once by regarding them as a single complex parameter $γ:= μ+ iσ$. The region should be a domain in the complex plane, and we will give a simple and concrete formula to give the region as a disc and a square.

math.ST↗
arXiv2022

A note on the $f$-divergences between multivariate location-scale families with either prescribed scale matrices or location parameters

We first extend the result of Ali and Silvey [Journal of the Royal Statistical Society: Series B, 28.1 (1966), 131-142] who first reported that any $f$-divergence between two isotropic multivariate Gaussian distributions amounts to a corresponding strictly increasing scalar function of their corresponding Mahalanobis distance. We report sufficient conditions on the standard probability density function generating a multivariate location family and the function generator $f$ in order to generalize this result. This property is useful in practice as it allows to compare exactly $f$-divergences between densities of these location families via their corresponding Mahalanobis distances, even when the $f$-divergences are not available in closed-form as it is the case, for example, for the Jensen-Shannon divergence or the total variation distance between densities of a normal location family. Second, we consider $f$-divergences between densities of multivariate scale families: We recall Ali and Silvey 's result that for normal scale families we get matrix spectral divergences, and we extend this result to densities of a scale family.

math.ST↗
arXiv2022

Information measures and geometry of the hyperbolic exponential families of Poincar\'e and hyperboloid distributions

We study various information-theoretic measures and the information geometry of the Poincar\'e distributions and the related hyperboloid distributions, and prove that their statistical mixture models are universal density estimators of smooth densities in hyperbolic spaces. The Poincar\'e and the hyperboloid distributions are two types of hyperbolic probability distributions defined using different models of hyperbolic geometry. Namely, the Poincar\'e distributions form a triparametric bivariate exponential family whose sample space is the hyperbolic Poincar\'e upper-half plane and natural parameter space is the open 3D convex cone of two-by-two positive-definite matrices. The family of hyperboloid distributions form another exponential family which has sample space the forward sheet of the two-sheeted unit hyperboloid modeling hyperbolic geometry. In the first part, we prove that all $f$-divergences between Poincar\'e distributions can be expressed using three canonical terms using Eaton's framework of maximal group invariance. We also show that the $f$-divergences between any two Poincar\'e distributions are asymmetric except when those distributions belong to a same leaf of a particular foliation of the parameter space. We report closed-form formula for the Fisher information matrix, the Shannon's differential entropy and the Kullback-Leibler divergence. and Bhattacharyya distances between such distributions using the framework of exponential families. In the second part, we state the corresponding results for the exponential family of hyperboloid distributions by highlighting a parameter correspondence between the Poincar\'e and the hyperboloid distributions. Finally, we describe a random generator to draw variates and present two Monte Carlo methods to stochastically estimate numerically $f$-divergences between hyperbolic distributions.

cs.IT↗
arXiv2022

Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution

We establish some limit theorems for quasi-arithmetic means of random variables. This class of means contains the arithmetic, geometric and harmonic means. Our feature is that the generators of quasi-arithmetic means are allowed to be complex-valued, which makes considerations for quasi-arithmetic means of random variables which could take negative values possible. Our motivation for the limit theorems is finding simple estimators of the parameters of the Cauchy distribution. By applying the limit theorems, we obtain some closed-form unbiased strongly-consistent estimators for the joint of the location and scale parameters of the Cauchy distribution, which are easy to compute and analyze.

math.ST↗
arXiv2021

On $f$-divergences between Cauchy distributions

We prove that the $f$-divergences between univariate Cauchy distributions are all symmetric, and can be expressed as strictly increasing scalar functions of the symmetric chi-squared divergence. We report the corresponding scalar functions for the total variation distance, the Kullback-Leibler divergence, the squared Hellinger divergence, and the Jensen-Shannon divergence among others. Next, we give conditions to expand the $f$-divergences as converging infinite series of higher-order power chi divergences, and illustrate the criterion for converging Taylor series expressing the $f$-divergences between Cauchy distributions. We then show that the symmetric property of $f$-divergences holds for multivariate location-scale families with prescribed matrix scales provided that the standard density is even which includes the cases of the multivariate normal and Cauchy families. However, the $f$-divergences between multivariate Cauchy densities with different scale matrices are shown asymmetric. Finally, we present several metrizations of $f$-divergences between univariate Cauchy distributions and further report geometric embedding properties of the Kullback-Leibler divergence.

cs.IT↗