2019年度年会(於:東京工業大学)

SUMMARY: Quantum walk is a quantum version of random walk and has been extensively studied since around 2000. A striking property of the quantum walk is the spreading property. The standard deviation of the walker’s position grows linearly in time, quadratically faster than random walk, i.e., ballistic spreading. On the other hand, a walker stays at the starting position: localization occurs. In this talk, as an autobiographical sketch of my life, I address a route to my work on quantum walks via my previous work on interacting particle systems. Therefore, my title is “From interacting particle systems to quantum walks”. Moreover, due to the rapid development of quantum computer by huge IT companies recently, it has become a reality that programs based on quantum walks run on quantum computers. Finally, I briefly explain the recent trends.

SUMMARY: As a generalization of Gaussian orthogonal/unitary/symplectic ensembles, Gaussian beta ensembles, one of the most studied models in random matrix theory, were originally defined in terms of the joint density of eigenvalues. They have been studied by using some methods in statistical mechanics since the distributions of eigenvalues can be viewed as the equilibrium measures of a one-dimensional Coulomb log-gas with an external Gaussian potential. Gaussian beta ensembles are now realized as eigenvalues of certain random tridiagonal matrices. Since the discovery of the random matrix models, many new spectral properties of Gaussian beta ensembles have been established. This talk gives a brief survey on recent developments with emphasizing on the global regime which deals with the convergence to a limiting measure, and the fluctuation around the limit of the empirical distributions.

SUMMARY: We consider statistical inference for high-dimension, low-sample-size (HDLSS) data. It is very important for HDLSS data that one selects a suitable procedure depending on the high-dimensional eigenstructures. Aoshima and Yata (2018, Sinica) proposed two eigenvalue models for high-dimensional data. One is called strongly spiked eigenvalue (SSE) model and the other one is called non-SSE (NSSE) model. A lot of theories and methodologies for HDLSS data have been developed under the NSSE model. In this talk, we focus on the SSE model that is often seen when we analyze microarray data sets. We give new theories and procedures under the SSE model. As for the SSE model, usually, one cannot discuss the asymptotic normality. In order to overcome this problem, Ishii, Yata and Aoshima (2016, JSPI) newly gave asymptotic distribution of the largest eigenvalue. On the other hand, Aoshima and Yata (2018, Sinica) gave the data-transformation technique that transforms the SSE model into the NSSE model. By using the high-dimensional asymptotics and the data-transformation technique, we construct new two sample test procedures, equality tests of two covariance matrices, classification procedures and so on. We also give numerical results of our new procedures and demonstrations by using microarray data sets.

SUMMARY: For the analysis of square contingency tables with the same row and column ordinal classifications, this presentation proposes new models, which may be appropriate for a square contingency table if it is reasonable to assume an underlying bivariate distribution. These models have characteristics that the cell probabilities have a similar structure of underlying bivariate distribution. The simulation studies based on some bivariate distributions are given.

SUMMARY: We show that weighted periodic points and iterated preimages of the Gauss map are qui-distributed according to the Gauss map. Our proof is based on the Large Deviation Principle and the uniqueness of the minimizer of the corresponding rate function.

SUMMARY: In this talk, we study the condition for normal numbers in numerical systems. Let \(b\) be an integer greater than 1 and \(r,s\) positive integers. Maxfield showed for any real number \(x\) that \(x\) is normal in base \(b^r\) if and only if \(x\) is normal in base \(b^s\). Recall that normal numbers in base \(b\) are denoted in terms of generic points of a certain dynamical system. The main purpose of this talk is to compare the generic points of two ergodic measure preserving systems. Using our main results, we obtain that the sets of the generic points in certain two different ergodic measure preserving systems coincide.

SUMMARY: We investigate recurrent and transient behavior for expanding maps on the real line. Our results provide a one-dimensional model for the phenomenon of dimension gaps which occur for limit sets of Kleinian groups. We use ergodic theory and in particular, thermodynamic formalism.

SUMMARY: We give some estimates for discrepancies of irrational rotations with several large partial quotients and report unusual aspects of the behavior of discrepancies caused by several large partial quotients.

SUMMARY: We discuss scaling limits for Glauber–Kawasaki dynamics. The Glauber–Kawasaki dynamics has been introduced by De Masi et al. to derive a reaction-diffusion equation from a microscopic particle system. In fact, they derived a reaction-diffusion equation as a limiting equation of the density of particles. This limit is usually called hydrodynamic limit. In this talk, we focus on several scaling limits related to this hydrodynamic limit.

SUMMARY: We consider the sequence of equilibrium measures for a given function parametrized by temperature. Temperature controls ordered and disordered powers, the potential function and entropy. The lower temperature the goes, the more the potential function effects strengthen. In this talk we pay attention to behavior of equilibrium measures as the temperature goes to zero. A fundamental problem in the zero temperature limit is the convergence of equilibrium measures. In the one-dimensional case, the sequence of equilibrium measures for a locally constant function converges. However in the high-dimensional case, there exists a locally constant function whose sequence of equilibrium measures does not converge. We construct such a locally constant function in dimension two by imbedding a one-dimensional effective subshift into a two-dimensional subshift of finite type.

SUMMARY: There are three typical random point fields with infinitely many particles in log-gases on 1-dimimension, that is, the Bessel, the Airy, and the sine random point fields. Furthermore there exists transition relations between the three random point fields. In this talk, we discuss dynamical version of the transition relations.

SUMMARY: Determinantal point processes (DPPs) appear in various models such as uniform spanning trees, uniform lozenge tilings, eigenvalues of random matrices. The former two are DPPs on discrete spaces, and the last is on continuum spaces. There are some interesting properties which are proved only in discrete cases. Tail triviality is one of them. We consider a DPP \( \mu \) on a continuum space \( S \) with an Hermitian symmetric kernel function \( K : S \times S \rightarrow \mathbb {C}\). We prove tail triviality of \( \mu \) by constructing tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure.

SUMMARY: We consider (not necessarily Markovian) continuous time random walks on Young diagrams as microscopic dynamics keeping the Plancherel measures invariant. We derive evolution of macroscopic profiles under diffusive scaling limit by using free probability and harmonic analysis on the symmetric group. Furthermore we illustrate an anomalous phenomenon observed with a pausing time obeying a heavy-tailed distribution without the mean.

SUMMARY: In this talk, we show homogenization of symmetric \(d\)-dimensional Lévy processes. Homogenization of one-dimensional pure jump Markov processes has been investigated by Tanaka et al.; their motivation was the work by Bensoussan et al. on the homogenization of diffusion processes in \({\mathbb R}^d\).

We investigate a similar problem for a class of symmetric pure-jump Lévy processes on \({\mathbb R}^d\) and we identify —using Mosco convergence— the limit process.

SUMMARY: General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces \((S, {\mathcal B}(S), \mu )\), with \(S\) weighted \(l^2\)-spaces, \({\mathcal B}(S)\) the Borel \(\sigma \)-field of \(S\), and \(\mu \) a Borel probability measure on \(S\), are introduced. A family of non-local Markovian symmetric forms \({\mathcal E}_{\alpha }\), \(0 < \alpha \leq 1\), acting in each given \(L^2(S; \mu )\) is defined, the index \(\alpha \) characterizing the order of the non-locality. It is shown that all the forms \({\mathcal E}_{\alpha }\) defined on \(\bigcup _{n \in {\mathbb N}} C^{\infty }_0({\mathbb R}^n)\) are closable in \(L^2(S;\mu )\), and sufficient conditions, under which the closure of the closable forms (Dirichlet forms), become quasi-regular, are given. Then, an existence theorem of \(\alpha \)-stable type Hunt processes properly associated to the Dirichlet forms is given. As an application of the theorems, the problem of stochastic quantizations of Euclidean \(\Phi ^4_3\)-fields by means of \(\alpha \)-stable type Hunt processes is discussed.

SUMMARY: In this talk, we study optimality conditions for quasiconvex programming in terms of subdifferentials. We show a necessary and sufficient optimality condition for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. We introduce a necessary and sufficient optimality condition for non-essentially quasiconvex programming in terms of Martínez-Legaz subdifferential. Additionally, we show a necessary optimality condition for quasiconvex programming with a reverse quasiconvex constraint in terms of Greenberg–Pierskalla subdifferential.

SUMMARY: In this study, we consider a Markov decision process model with a converging branch system which is one of the nonserial transition systems. We introduce recursive equations by using dynamic programming technique.

SUMMARY: We propose the Newton–Kantorovitch method for solving partially coupled forward-backward stochastic differential equations (FBSDEs) involving smooth coefficients with uniformly bounded derivatives. We show the global convergence property with respect to \(T>0\), moreover, it is quadratic convergence.

SUMMARY: Coherent risk measures in financial management are discussed from the view point of average value-at-risks with risk spectra. A minimization problem of the distance between risk estimations through decision maker’s utility and coherent risk measures with risk spectra is introduced. The risk spectrum of the optimal coherent risk measures in this problem is obtained and it inherits the risk averse property of utility functions. Various properties of coherent risk measures and risk spectrum are demonstrated.

SUMMARY: In this talk, we first consider Malliavin differentiability of indicator functions on canonical Lévy spaces. By using it, we obtain explicit representations of locally risk-minimizing hedging strategy for digital options in markets driven Lévy processes.

SUMMARY: Binary time series is the time series converted into 0 and 1. In this talk, a strictly stationary ellipsoidal alpha-mixing bivariate process with mean zero and finite variance is discussed. We consider the estimation problems of the functional spectra of a bivariate time series by using binary time series. First, we show the consistency of our estimator. Next, we elucidate the joint asymptotic distribution of our estimator.

SUMMARY: It is the fundamental task of statistics to find out internal relationship of diversity of scientific observations. Quantile regression offers the opportunity for a more complete view of the relationships among stochastic variables. In this talk, the properties of modified LASSO estimators for linear quantile regression models is discussed when the disturbances are long-memory which implies the dependence on the disturbances before decays very slowly. We derive the asymptotic distributions of the estimators when there is no nonzero parameters and also derive the property of the estimators when nonzero parameters exist under some appropriate regularity conditions.

SUMMARY: Analysis of variance (ANOVA) is tailored for independent observations. Recently, there has been considerable demand for the ANOVA of high-dimensional and dependent observations in many fields. Thus, it is important to analyze the differences among big data’s averages of areas from all over the world, such as the financial and manufacturing industries. However, the numerical accuracy of ANOVA for such observations has been inadequately developed. Thus, herein, we study the Edgeworth expansion of distribution of ANOVA tests for high-dimensional and dependent observations. Specifically, we present the second-order approximation of classical test statistics proposed for independent observations. We also provide numerical examples for simulated high-dimensional time-series data.

SUMMARY: We consider a test for missingness in time series. Suppose we observe a time series with missing values, which is generated by a regression model with dependent disturbances. The mechanism for the missing values is supposed to be generated by Bernoulli responses from a generalized linear ARMA model. We propose a test statistic for a score type test for the null hypothesis that the data are missing completely at random. For this testing problem, we use the Laplace approximation to obtain the likelihood of the process. We investigate the performance of our proposed test statistic in several numerical simulations. The method is also applied to real data of pollution levels containing some missing observations.

SUMMARY: Statistical treatment of a circular observation has attracted much attention in these decades, and such data is often observed in variety of fields. This talk constructs an \(L_1\)-based local polynomial regression estimator for a nonlinear regression function of circular random variables. We use a circular kernel to approximate the regression function by a polynomial function locally. The novel aspect of this talk is that we allow the dependent structure and possibly infinite variance of the error process. The result in Di Marzio, Panzera and Taylor (2009) is then nicely extended to infinite variance dependent innovation case. Some simulation experiments illustrate the finite sample performance of the proposed method and elucidate robustness of the proposed \(L_1\)-based estimator.

SUMMARY: Weak convergences of marked empirical processes in \(L^2(\mathbb {R},\nu )\) and their applications to statistical goodness-of-fit tests are provided, where \(L^2(\mathbb {R},\nu )\) is the set of equivalence classes of the square integrable functions on \(\mathbb {R}\) with respect to a finite Borel measure \(\nu \). The results obtained in our framework of weak convergences are, in the topological sense, weaker than those in previous works. However, our results have the following merits: (1) avoiding conditions which do not suit for our purpose; (2) treating a weight function which make us possible to propose an Anderson–Darling type test statistics for goodness-of-fit tests. Indeed, applications are novel.

SUMMARY: In this talk, we consider testing high-dimensional covariance structures: (i) diagonal matrix and (ii) intraclass covariance matrix. We produce a test statistic for each covariance structure by using the extended cross-data-matrix (ECDM) methodology and show the unbiasedness of the ECDM test statistic even in a high-dimensional setting. We also show that the ECDM test statistics have a consistency property and hold the asymptotic normality. We propose a new test procedure based on the ECDM test statistic for each hypothesis and evaluate its size and power asymptotically.

SUMMARY: There are many versions of Bayesian Cramér–Rao type lower bounds of the Bayes risk. We compare them from the point of view of asymptotic optimality. We show that the asymptotic optimality result of Abu-Shanab and Veretennikov (2015) still holds true in the sense of Bhattacharyya type lower bound of Koike (2006) in univariate case. And we show the asymptotic optimal choice in the lower bound of Gill and Levit (1996) in multivariate case.

SUMMARY: Ćwik and Mielniczuk (1989) suggested a nonparametric direct density ratio estimator based on the kernel density estimator. However, their direct density ratio estimator is inconsistent near the boundary, similarly to the kernel density estimator. In this talk, the asymptotic properties of a direct density ratio estimator based on the beta kernel density estimator, which is free of boundary bias, are studied.

SUMMARY: Asymmetric kernel density estimation has been well-studied in the literature. In this talk, extending several bias reduction methods with \(n^{-8/9}\)-MISE, it is shown that some asymmetric kernel density estimators can be easily bias-corrected up to the higher-order, in an additive or multiplicative way. We prove that new higher-order bias-corrected asymmetric kernel density estimators have the desirable asymptotic properties under suitable conditions.

SUMMARY: In this talk, we propose a new Kolmogorov–Smirnov type test which is based on kernel estimation. The new test statistics is based on a boundary-free kernel test. We discuss asymptotic properties of the statistic and compare powers of the ordinal and new Kolmogorov–Smirnov tests by simulation.

SUMMARY: We shall propose a new measure of symmetry for square contingency tables having ordered categories. The measure is expressed as a weighted geometric mean of the diversity index. The proposed measure is useful for comparing the degrees of departure from partial symmetry between two different ordinal tables.

SUMMARY: We propose a new measure of marginal homogeneity for square contingency tables with ordered categories. The measure is expressed as a weighted geometric mean of the diversity index. In this talk,we show some properties of the proposed measure and give its confidence interval.

SUMMARY: For multi-way contingency tables with ordered categories, we are interested in considering the marginal homogeneity model which indicates the structure of equality of marginal distributions (Agresti, 2002, p. 440; Bhapkar and Darroch, 1990). When the marginal homogeneity model does not fit for the data, we are also interested in seeing the structure of inhomogeneity of marginal distributions. So, some extensions of the marginal homogeneity model were proposed. This presentation proposes two models using the complementary log-log transform. It also gives the decompositions of the marginal homogeneity model into the proposed model and a model of the equality of marginal means.

SUMMARY: For the analysis of square contingency table, Yoshimoto, Tahata, Iki and Tomizawa (2018) considered the covariance symmetry model and pointed out that the symmetry model holds if and only if both the covariance symmetry model and the marginal homogeneity model hold. This presentation proposes the moment symmetry model and the decomposition theorem of the marginal symmetry and symmetry models for multi-way contingency tables. The moment symmetry model and decomposition theorem are the generalization of the result given by Yoshimoto et al. (2018).

SUMMARY: For the analysis of multidimensional contingency tables with ordinal categories, our interest is whether each marginal distribution is homogeneous or not. It may be more appropriate to apply a certain marginal inhomogeneity model when the marginal homogeneity model does not fit. In this report, we propose the generalized marginal inhomogeneity model using a continuous strictly increasing function. Using this model, we prove that the marginal homogeneity model is decomposed into two models. The decomposition is useful to deduce the reason for the poor fit when the marginal homogeneity model fits poorly.

SUMMARY: We consider the testing problem for a mean vector when the data matrix is of the monotone missing pattern. The simplified \(T^2\)-type test statistic for this problem has been derived by Krishnamoorthy and Pannala (1999) and Yagi et al. (2018). Further, its null distribution was given in the form of an asymptotic expansion and the transformation for the test statistic was derived by Yagi et al. (2018). In this talk, we propose a new test statistic designed based on the above result. Further, we present the approximation to the upper percentiles of this statistic and propose the transformed test statistics. Finally, by a Monte Carlo simulation, we investigate the accuracy and asymptotic behavior of the approximation for \(\chi ^2\) distribution.

SUMMARY: In three dimensional contingency tables, test of complete independence, test of independence between one factor and the other two and test of conditional independence are important and interesting problems. In this report, we consider improvement of usual test statistics of above independencies by using Bartlett-type transformation and improved transformation.

SUMMARY: Event-related functional Magnetic Resonance Imaging (efMRI) enables us to estimate the peak of the hemodynamic response function (HRF), describing changes in the blood oxygen level dependent (BOLD) to neural activity in response to mental stimuli. As a good candidate of arrays providing highly efficient designs of stimuli, Lin–Phoa–Kao (2017) introduced the concept of circulant almost orthogonal array (CAOA). Yoshida–Satake–Phoa–Sawa (2018) gave a systematic construction of CAOA with strength 3 by using cyclic Hadamard 2-designs. When we construct CAOA by using Paley’s 2-designs, we must find suitable subsequences in the characteristic sequence of quadratic residues modulo primes. In this talk, we investigate the location of such subsequences.

SUMMARY: In this talk we are mainly concerned with a class of cubature formulas with respect to rotationally invariant bivariate integrals, which has been traditionally studied in numerical analysis, combinatorics and design of experiments. The aim of this talk is to mention an algebraic motivation for a certain result recently obtained as a joint work with Masatake Hirao.

SUMMARY: Dropout is used in deep learning. It is a method of learning by invalidating nodes with randomly for each layer in the multi-layer neural network. And it deletes a random sample of activations (nodes) to zero during the training process. A random sample of nodes cause more irregular frequency of dropout edges. A dropout design is a combinatorial design on dropout nodes from each partite which balances frequency of edges. In this talk, we give some constructions of dropout designs using projective spaces or affine spaces.

SUMMARY: The concept of a splitting-balanced block design (SBD) has been defined with some applications for authentication codes in Ogata et al. (2004). Most of the SBDs obtained in literature contain many repeated subblocks. From a point of view for some applications it is preferable that the design here has no repeated subblocks. In this talk, a splitting-balanced packing-block design (SPD) is newly defined, and also direct and recursive constructions of a \((v,2\times 2,1)\)-SPD with a cyclic automorphism are provided. It is finally shown that there exists a cyclic \(((2P-1)Q,2\times 2,1)\)-SPD, where \(P\) is any product of primes \(p\) with each \(p\equiv 1\) (mod 4) and \(Q\) is any product of primes \(q\) with each \(q\equiv 1\) (mod 8).

SUMMARY: We consider the third-order linear model for \(3^{m}\) factorials. Let \(T\) be a \(3^{m}\)-BTO design of resolution \(\mathrm {R}^{\ast }(\{10,01\})\) derived from an \(\mathrm {SA}(m;\{\lambda _{xm-x-yy}\})\) with \(N\) assemblies and \(m\ge 6.\) Further let \(\sigma ^{4m}D_{T}\) be the determinant of the variance-covariance matrix of the estimators concerning with all the main effects based on \(T\). If \(D_{T}\le D_{T^{\ast }}\) for any \(T^{\ast },\) then \(T\) is said to be D\(^{\ast }\)-optimal, where \(T^{\ast }\) is a \(3^{m}\)-BTO design of resolution \(\mathrm {R}^{\ast }(\{10,01\})\) derived from an SA with \(N\) assemblies. In this talk, we give D\(^{\ast }\)-optimal \(3^{m}\)-BTO designs of resolution \(\mathrm {R}^{\ast }(\{10,01\})\) derived from SA’s for \(6\le m\le 8,\) where \(N<\nu (m).\) Here \(\nu (m)\) is the number of non-negligible factorial effects.

SUMMARY: The concept of spherical design is introduced by Delsarte–Geothals–Seidel (1977). There exist several works on spherical design, e.g., the relationships of experimental design or cording theory. However, as far as the authors know, there exist few studies on spherical-cap cases. In this talk, the new concept of spherical-cap design is firstly introduced as a generalization of spherical design. Secondary, the lower bounds of number of points in a spherical-cap design and tight spherical-cap design, which attains such a lower bound, are discussed. Finally, several examples of spherical-cap designs are presented and some applications of such designs are discussed if possible.

SUMMARY: The concept of almost tight Euclidean design, which is a union of the origin and a tight design satisfying \(0 \not \in X\), is introduced by Bannai et al (2010). As far as the authors know, there exist few studies on characterization of such designs for circularly symmetric integrals. In this talk we give a necessary condition for the existence of such designs in terms of near Gaussian quadrature formulas for the radial component of a circularly symmetric integral. In particular, we focus on four types of Jacobi weights and give several examples of such designs.

SUMMARY: Spherical designs are “good” point sets which give exact numerical integration rules for spherical polynomials. There exist many works on how to construct such designs. For example, Yamamoto et al (2018) discusses a method of constructing such designs using the hyperoctahedral group orbits of corner vencors and the internally dividing points and presents such designs. In this talk, we give a generalization of their method and several examples of such designs. In particular, we note that the list of known minimum points of spherical design has been improved in some examples.