2019年度秋季総合分科会(於:金沢大学)

SUMMARY: We discuss in this talk recent progress on scaling limits for exclusion processes. In particular, this talk shall focus on some sort of law of large numbers for the empirical measure of the particle system, which is often referred to as hydrodynamic limit. Corresponding fluctuations and large deviations are also discussed. The scaling limits for additive functionals and a tagged particle are also mentioned.

SUMMARY: In describing properties of disordered media, physicists have long been interested in the behaviour of random walks on random graphs that arise in statistical mechanics, such as percolation clusters and various models of random trees. Random walks on random graphs are also of interest to computer scientists in studies of complex networks. In ‘critical’ regimes, many of the canonical models exhibit large-scale fractal behaviour, which mean it is often a challenge to describe their geometric properties, let alone the associated random walks. However, in recent years, the deep connections between electrical networks and stochastic processes have been advanced so that tackling some of the key examples of random walks on random graphs is now within reach. In this talk, I will introduce some recent work in this direction, and describe some prospects for future developments.

SUMMARY: A method that uses order statistics to construct multivariate distributions with fixed marginals is proposed by Baker (2008). We investigate the properties of Baker’s bivariate distributions. The properties include the weak convergence to the Fréchet–Hoeffding upper bound, the product-moment convergence and the totally positivity of order 2. As Baker’s distribution utilizes a representation of the Bernstein copula in terms of a finite mixture distribution, we propose expectation-maximization (EM) algorithms to estimate the Bernstein copula and give illustrative examples using real data sets and a 3-dimensional simulated data set. These studies show that the Bernstein copula is able to represent various distributions flexibly and that the proposed EM algorithms work well for such data.

Using B-spline functions, we construct a new class of copulas, B-spline copulas, that includes the Bernstein copulas as a special case. The range of correlation of the B-spline copulas is examined, and the Fréchet–Hoeffding upper bound is proved to be attained when the number of B-spline functions goes to infinity. As the B-spline functions are well-known to be an order-complete weak Tchebycheff system, we show that the property of total positivity of any order (TP\({}_\infty \)) follows for the maximum correlation case. These results extend the classical results for the Bernstein copulas. In addition, we derive in terms of the Stirling numbers of the second kind an explicit formula for the moments of the related B-spline functions on \([0,\infty )\).

SUMMARY: Bayesian inference under model misspecification has been developed in recent years. In such cases, the usual Bayesian updating is meaningless, and one of the strategies is the use of the general posterior distributions based on the general Bayesian updating. In this talk, we give an overview of this framework, and talk about applications to robust statistics.

SUMMARY: When a Brownian motion is given as underlying process and a stable random measure is given as basis of continuous additive functional for locally admissible branching rate functional, then we can construct a superprocess with those data and the initial measure. We discuss the corresponding historical process and give an estimate on good paths of the historical superprocess.

SUMMARY: We establish the multifractal formalism for the generalised local dimension spectrum of a Gibbs measure \(\mu \) supported on the attractor \(\Lambda \) of a conformal iterated functions system on the real line. Namely, for \(\alpha \in R\), we prove a formula for the Hausdorff dimension of the set of \(x\in \Lambda \) for which the \(\mu \)-measure of a ball of radius \(r_{n}\) centred at \(x\) obeys a power law \(r_{n}{}^{\alpha }\), for a sequence \(r_{n}\rightarrow 0\).

SUMMARY: Recently, some researchers have started to study the limit sets (for short, fractals) generated by non-autonomous iterated function systems (for short, NAIFSs). However, the NAIFSs in such studies are genereted by the functions defined on some compact set, which deduces that the fractals are always uniformly bounded with respect to the base points. In this talk, we consider the NAIFSs genereted by the functions defined on a complete metric space and we construct the fractals (which are not uniformly bounded with respect to the base points in general) generated by the NAIFSs. In addition, we discuss some basic properties of the fractals.

SUMMARY: In this talk, we focus on interval maps with two or more indifferent fixed points, and present a strong distributional limit theorem for the joint-law of the occupation times for neighborhoods of indifferent fixed points. The scaling limit is a multidimensional version of Lamperti’s generalized arcsine distribution, which is the joint-law of occupation times of a skew Bessel diffusion processes moving on multiray.

SUMMARY: In this talk, we shall consider a generalization of the Loewner equation to parallel slit half-planes. This equation describes the evolution of conformal mappings and, these days, is paid much attention due to the great success of Schramm–Loewner evolution (SLE). SLE is now extended toward two directions: One direction is to consider multiple SLE paths simultaneously, and the other is to consider SLE on multiply connected domains. In view of the modern Loewner theory in complex analysis, we shall discuss a framework broad enough to include both the directions by establishing the Komatu–Loewner equation with measure-valued driving sources. One of our key tools is Brownian motion with darning (BMD).

SUMMARY: In this talk, we consider the so-called frog model with random initial configurations, which is described by the following evolution mechanism of simple random walks on the multidimensional cubic lattice: Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and they independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start doing independent simple random walks. We observe how initial configurations affect the asymptotic shape of the set of all sites visited by active particles up to a certain time, and present the continuity for the asymptotic shape in the law of the initial configuration.

SUMMARY: Finding a ground state of a given Hamiltonian is an important. One of the potential methods is to use a Markov chain Monte Carlo (MCMC) to sample the Gibbs distribution whose highest peaks correspond to the ground states. In this talk, we use stochastic cellular automata (SCA) and see if it is possible to find a ground state faster than the Glauber dynamics. We show that, if the temperature is sufficiently high, it is possible for SCA to have more spin-flips per update in average than Glauber and, at the same time, to have an equilibrium distribution “close” to the Gibbs distribution. We also propose a new way to characterize how close a probability measure is to the target Gibbs.

SUMMARY: PCOC (pronounced as peacock) is an acronym for French words \(P\)rocessus \(C\)roissant pour l’\(O\)rdre \(C\)onvexe, words for an integrable process which is increasing in the convex order. In this presentation, we prove that the time-average of exponential of a fractional Brownian motion is a PCOC.

SUMMARY: The present paper establishes a discrete version of the result obtained by P. Carr and S. Nadtochiy for 1-dimensional diffusion processes. Our result is for Markov chains on multi-dimmensional lattice.

SUMMARY: We study the behavior of a class of edge-reinforced random walks on the half-line, with heterogeneous initial weights, where each edge weight can be updated only when the edge is traversed from left to right. We provide a description for different behaviors of this process and describe phase transitions that arise as trade-offs between the strength of the reinforcement and that of the initial weights. Our result aims to complete the ones given by Davis (1989, 1990), Takeshima (2000, 2001), and Vervoort (2000).

SUMMARY: We studied the bail-out optimal dividend problem with regime switching under the constraint that the cumulative dividend strategy is absolutely continuous. We confirm the optimality of the regime-modulated refraction-reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. To verify the conjecture of a barrier type optimal control, we first introduce and study an auxiliary problem with the final payoff at an exponential terminal time. Second, we transform the problem with regime-switching into an equivalent local optimization problem with a final payoff up to the first regime switching time. The refraction-reflection strategy with regime-modulated thresholds can be shown as optimal by using results in the first step and some fixed point arguments for auxiliary recursive iterations.

SUMMARY: We consider the diffusion approximation of a G/Ph/n queue with customer abandonment in the Halfin–Whitt heavy-traffic regime and extend the conventional locally Lipschitz hazard-type scaling of abandonment distribution to a more general scaling under which not only the non-locally Lipschitz hazard-type case but also a wider range of abandonment distributions can be treated. For our objective, we first show the C-tightness of scaled customer-count processes and then prove that the stochastic equation satisfied by any limit process has the uniqueness in law of the solution by applying the Girsanov transformation to the localized equation.

SUMMARY: In this talk, we give a definition of unimodality to discrete distributions on the real line according to a modification of strong unimodality on the lattice. Then, we can also consider a definition of linear unimodal for discrete distributions, where one generalization of linear independence over the rationals plays an important role to linear combinations of discrete valued random variables.

SUMMARY: In this talk we introduce the notion of Perron complements of Ruelle operators and investigate its some properties. This notion was first given to nonnegative matrices by Meyer in [Lin. Alg. Appl., 114, 69–94 (1989)]. We extend his notion to Perron complements of operators. The complements we give have properties similar to the original complements: a hereditary property and decomposition theorems of eigenvectors and of Gibbs measures. Our results are useful for applications in a system with holes.

SUMMARY: The aim of the talk is twofold: first, we intend to present a simple but nontrivial example of the numerical integration whose precision level exceeds the limit \(O(n^{-1})\), second we like to emphasize that this new scheme is constructed in the framework of the noncausal stochastic calculus introduced by the author in 1979.

SUMMARY: One-dimensional Random Walk in Cooling Random Environment (RWCRE) is obtained by starting from one-dimensional Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times. In this talk, we focus on how the recurrence versus transience criterion known for RWRE changes for RWCRE and explore the fluctuations for RWCRE when RWRE is either recurrent or satisfies a classical central limit theorem. An “overarching” goal of this topic is to investigate how the behaviour of a random process with a rich correlation structure can be affected by re-settings.

SUMMARY: On disconnected self-similar fractal sets, random processes can be defined as limit of suitably scaled random walks. The scaled random walks lead to a super-diffusion, that is, the diffusion exponent is larger than one. In this talk, we consider Brox’s diffusion processes on the sets, which move much slower than the random processes influenced by random environments which are independent from the random processes.

SUMMARY: In this talk, we study spectral properties of explosive symmetric Markov processes. Under a condition on its life time, we prove the \(L^1\)-semigroup of Markov processes become compact operators.

SUMMARY: Consider the Schrödinger form with a perturbation which consists of two measures. We establish the precise asymptotic behavior of the spectral function for the Schrödinger form. This result extends the preceding one by Nishimori which treated the differentiability of the spectral function.

SUMMARY: In this talk, we determine the long time behavior and the exact order of the tail probability for the maximal displacement of a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of the associated Schrödinger type operator. We also prove the existence of the Yaglom type limit for the distribution of the population outside the forefront. To establish our results, we show a sharp and locally uniform growth order of the Feynman–Kac semigroup.

SUMMARY: Heath–Jarrow–Morton model is the most general model of interest rate in mathematical finance. Musiela (1993) derived that this model reduces to a stochastic partial differential equations (SPDE). This SPDE is called by Heath–Jarrow–Morton–Musiela (HJMM) equation. In this talk, we consider the existence and uniqueness of the solution of HJMM equation driven by Lévy noise. Kusuoka (2000) showed the existence and uniqueness of weak solution of this equation under the boundary condition in the case of Wiener process. We extend this approach to Lévy Process.

SUMMARY: Motivated from time-inconsistent stochastic control problems, we introduce a new type of coupled forward-backward stochastic systems, namely, flows of forward-backward stochastic differential equations. They are systems consisting of a single forward stochastic differential equation (SDE) and a continuum of backward SDEs (BSDEs), which are defined on different time intervals and connected via an equilibrium condition. We formulate a notion of equilibrium solutions in a general framework and discuss the well-posedness of the equations.

SUMMARY: An explicit martingale representation for random variables described as a functional of a Lévy process will be given. The Clark–Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Itô’s formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Lévy models. Since the payoff of digital options is described by an indicator function, we also discuss the Malliavin differentiability of indicator functions with respect to Lévy processes.

SUMMARY: In this presentation, we formulate the Bayesian Markov decision processes with disturbances. We introduce the Bayesian approach to investigate decision process with disturbances where the transition probability depends on some parameter.

SUMMARY: The Laplace spectral density kernels are a new type of spectral density, which characterize the collection of all marginal bivariate distribution in a given stationary time series, in the absence of moment assumptions. In this talk, we consider a Kolmogorov–Smirnov (KS) test for Laplace spectral density kernels. This test, thus, is a goodness-of-fit test for the collection of all bivariate marginals of an observed series. First, we derive the asymptotic null distribution of the KS statistic which, however, is not distribution-free. We therefore propose a numerical method, combined with the estimation of a covariance kernel, for the computation of critical values. Finally, we show that our testing procedure is consistent.

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 asymptotic properties of modified LASSO estimators for linear quantile regression models are developed, when the disturbances are long-memory which implies the dependence on the disturbances before decays very slowly, and when the dimension of regressor \(p\) varies with respect to the observation length \(n\). Especially, when \(p\) increases as \(n\) increases, it corresponds to a high-dimensional case.

SUMMARY: In this talk, we treat a model in which the finite variation part of a two-dimensional semi-martingale is expressed by time-integration of latent processes. We propose a correlation estimator between the latent processes and show its consistency and asymptotic mixed normality. Moreover, we propose two types of estimators for asymptotic variance of the correlation estimator and show their consistency in a high frequency setting. Our model includes doubly stochastic Poisson processes whose intensity processes are correlated Itô processes.

SUMMARY: We study the polynomial-type large deviation inequalities for quasi-likelihood functions for discretely and noisily observed diffusion processes by applying the results in Yoshida (2011, AISM). The inequalities lead to the mathematical validity of the adaptive Bayes-type estimators with the same asymptotic distributions as adaptive maximum-likelihood-type estimators in Nakakita and Uchida (2019, SJS). Furthermore, it is shown that both the adaptive maximum-likelihood-type estimators and the adaptive Bayes-type estimators have the convergence of moments. We also examine the behaviours of adaptive Bayes-type estimators in computational simulation, and check that their performance is indeed equivalent to that of the adaptive maximum-likelihood-type estimators.

SUMMARY: The maximum composite likelihood estimator parametric models of determinantal point processes will be discussed. Since the joint intensities of these point processes are given by determinant of positive definite kernels, we have the explicit form of the joint intensities for every order. This fact enables us to consider the generalized maximum composite likelihood estimator for every order. In this talk, the two step generalized composite likelihood estimator will be introduced. Moreover, the moment convergence of the estimator will be proved for stationary case.

SUMMARY: Let \(K(=K_{n,\theta })\) be a positive integer-valued random variable whose distribution is given by \({\rm P}(K = x) = \bar {s}(n,x) \theta ^x/(\theta )_n\) \((x=1,\ldots ,n) \), where \(\theta \) is a positive number, \(n\) is a positive integer, \((\theta )_n=\theta (\theta +1)\cdots (\theta +n-1)\), and \(\bar {s}(n,x)\) is the coefficient of \(\theta ^x\) in \((\theta )_n\) for \(x=1,\ldots ,n\). This formula describes the distribution of the length of a Ewens partition. As \(n\) tends to infinity, \(K\) asymptotically follows a normal distribution. Moreover, as \(n\) and \(\theta \) simultaneously tend to infinity, if \(n^2/\theta \to \infty \), \(K\) also asymptotically follows a normal distribution. In this presentation, error bounds for the normal approximation are provided. The result shows that the decay rate of the error changes due to asymptotic regimes.

SUMMARY: For the data supported on \([0,\infty )\) or \([0,1]\), asymmetric kernel density estimation has been well-studied in the recent literature. Such a density estimator is non-recursive, by construction. In this talk, we consider its recursive version and then discuss some desirable asymptotic properties under suitable conditions.

SUMMARY: In this talk, we discuss the nonparametric estimation of the hazard function when the data has covariates. After removing the effect of the covariates, we estimate the baseline hazard function, using kernel type estimators. We also obtain asymptotic mean squared error of the hazard function estimator.

SUMMARY: The space of probability density functions on the Euclidean space is decomposed into orbits with respect to coordinate-wise transformations. By Sklar’s theorem, each orbit has a unique copula density. In this research, we consider a problem of whether similar results hold for exponential-type distributions instead of copulas. It is shown that, under regularity conditions, the existence holds whereas the uniqueness fails.

SUMMARY: Consider discrete multivariate probability models where some parameters from the first sample and those from the second sample are proportional. In two sample problems under the the null hypothesis where the samples are from the same population is tested against the hypothesis where the samples are from the different population. It is found that the total information is equal to the sum of the within information and the between information in some case, but not equal in several cases. To investigate this phenomenon, we found some interesting results. Relationships between the above problems and the Fisher Information are also discussed.

SUMMARY: We consider AIC for selecting the degree in a growth curve model when the data set has a three-step monotone missing pattern. Throughout this talk, we assume that the data are missing completely at random (MCAR). In this talk, we prove that the AIC is an exact unbiased estimator of the AIC-type risk function defined by the expected log-predictive likelihood when the maximum likelihood estimator of unknown mean parameter vector with known covariance matrix is used.

SUMMARY: Bartlett–Nanda–Pillai test is one of the famous test for the linear hypothesis about the regression coefficients of the multivariate linear model. Under normality assumption, the null distribution function can be represented as an integral of a matrix beta function on some region. Using Laplace’s approximation method for integrals, an asymptotic expansion formula of the null distribution function is derived under a large sample and high dimensional asymptotic framework. An error bound for the derived approximation formula is also derived.

SUMMARY: In this talk, we consider a two-sample test for high-dimensional data. 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. Ishii (2017, HMJ) considered uni-SSE (USSE) and gave a two-sample test procedure by using the noise-reduction method given by Yata and Aoshima (2012, JMVA). However, Ishii (2017, HMJ) assumed the equality of the first eigenspaces. In this talk, we give a new test procedure without assuming the condition. We also give numerical results of our new test procedure and data analysis by using microarray data sets.

SUMMARY: In this talk, we consider clustering based on principal component analysis (PCA) for high-dimensional mixture data. First, we derive a geometric representation of high-dimension, low-sample-size (HDLSS) data taken from a mixture model. With the help of the geometric representation, we give geometric consistency properties of sample principal component scores in the HDLSS context. We show that PCA can cluster HDLSS data under certain conditions in a surprisingly explicit way. Finally, we demonstrate the performance of the clustering by using gene expression data sets.