2018年度年会(於:東京大学)

SUMMARY: We consider irreducible, strong Feller, symmetric Markov processes with tightness property, and call the class of symmetric Markov processes having such properties Class (T). If a symmetric Markov process in Class (T) is conservative, then it has a very strong recurrence property, and if not, then it explodes very fast. Using this fact, we can derive some spectral properties, for example, \(p\)-independence of the growth bound of its semi-group, compactness of semi-group, bounded continuity of eigenfunctions. As an application of these properties, we can prove the existence and uniqueness of quasi-stationary distribution.

By checking time-changed processes to be in Class (T), we give an analytic characterization of the criticality (or subcriticality) for Schroedinger forms. Moreover, we show that Green function for Schroedinger-type operator satisfies some principles in potential theory (eg. Ugaheri’s maximum principle, the continuity principle) if the principal eigenvalue of the time-changed process is greater than one.

SUMMARY: The first fundamental theorem of asset pricing roughly states that a stochastic process satisfies a property concerning the Itô integrals with respect to it, called the no-arbitrage property, if and only if the process has an equivalent martingale measure. The theorem for discrete-time finite-dimensional processes on finite probability spaces, which goes back to Harrison and Kreps (1979), is a restatement of Stiemke’s lemma of finite-dimensional linear algebra. The theorem in the general setting can thus be viewed as an infinite-dimensional extension of Stiemke’s lemma. In the continuous-time setting, there is more than one way to formulate the “no-arbitrage property.” In this presentation, an emphasis will be put on the \(L^0\) space of random variables and the technique of numeraire change.

SUMMARY: A set of \(\ell \) balanced incomplete block (BIB) designs with common parameters having pairwise additivity is called an \(\ell \)-pairwise additive BIB design. The \(\ell \)-pairwise additivity of BIB designs can be regarded as a decomposition of a BIB design and/or a composition of several BIB designs, and further yields BIB designs with \(\ell \) distinct block sizes. In this talk, we discuss comprehensively the existence of \(\ell \)-pairwise additive BIB designs through the relationship with other combinatorial structures. Especially, as related combinatorial structures, we focus on nested BIB designs, orthogonal arrays (mutually orthogonal latin squares), perpendicular arrays, difference matrices and finite geometries. Furthermore, we present some results on splitting-balanced block designs named newly. The pairwise additivity also relates to the splitting-balanced property. The splitting-balanced block design (also called a splitting BIB design in literature) was introduced for applications to authentication codes by Ogata et al. in 2004. Finally, the bound of the number of blocks and some existence of splitting-balanced block designs are provided.

SUMMARY: In this presentation, weak convergences of random processes taking values in a separable Hilbert space are discussed. Especially, by using a common approach based on the limit theory in \(L^2\) spaces, we study partial sum processes of dependent random variables which appear in the following two topics: (i) statistical change point testing; (ii) functional central limit theorem for logarithmic combinatorial assemblies. (i) Statistical change point testing. When there exist chronologically obtained data, it is in interest to determine whether there are structural changes or not. In the domain of statistics, tests for such hypothesis is considered in the so-called change point testing. For parametric change point problems, several works have proposed test procedures based on score functions. Following these works, we propose a procedure based on the functional of a weighted random process whose weak convergence is discussed in \(L^2(0,1)\), which is a modification of test statistics in previous works. (ii) Functional central limit theorem for logarithmic combinatorial assemblies. Assemblies are a class of random combinatorial structures which includes random permutations, random mappings, random forests of labeled trees, and so on. The law of component counts of assemblies is provided from independent Poisson variables combined with the conditioning relation. In previous works, the weak convergence of the partial sum process of component counts of logarithmic assemblies, which are assemblies satisfying the logarithmic condition, has been considered in the Skorokhod space. On the other hand, we consider the partial sum process involving a weight function, which makes the limit different, and derive its weak convergence in \(L^2(0,1)\).

SUMMARY: We consider a level-dependent Lévy process that changes its drift depending on the position of the process. We first generalize the refracted Lévy process of Kyprianou and Loeffen (2010), which changes its drift above a given threshold, to the multi-refracted case. We then extend the results for more general level-dependent Lévy processes. We show how fluctuation identities of these processes can be expressed via scale functions.

SUMMARY: We study the optimal periodic dividend problem where dividend payments can only be made at the jump times of an independent Poisson process.

SUMMARY: We are concerned with upper rate functions for a symmetric jump process on the Euclidean space generated by a regular Dirichlet form. We give a condition on the jumping function for the process to enjoy upper rate functions of Brownian motion type. Our condition implies that the second moment of the jumping function is finite.

SUMMARY: We prove that the (classical) normal distributions are freely selfdecomposable. More generally it is established that the Askey–Wimp–Kerov distributions are freely self-decomposable. The main ingredient in the proof is a general characterization of the freely selfdecomposable distributions in terms of the derivative of their free cumulant transform.

SUMMARY: In this talk, we consider the maxima of payoffs for the generalized St. Petersburg game. The maxima for the original St. Petersburg game cannot be normalized to converge to a nondegenerate limit distribution. However, tuning the parameters appearing in the generalization, we show the normalized maxima converge to the Fréchet distribution.

SUMMARY: We study a smooth unimodal map whose critical point is non-recurrent and flat. Assuming the critical order is either polynomial or logarithmic, we establish the large deviation principle and give a partial description of the zeroes of the corresponding rate function.

SUMMARY: In this talk, we consider continuous maps of a metric space and introduce relations between mixing properties such as decay of correlations and topologically mixing and Devaney’s chaos.

SUMMARY: Every factor map between given ergodic, measure-preserving transformations on infinite Lebesgue spaces can be realized as a proper, factor map between strictly ergodic, locally compact Cantor systems. A locally compact Cantor system is a topological dynamical system of a homeomorphism on a locally compact (non-compact) metric space, whose one-point compactification is a Cantor set. Such a system, or a homeomorphism, is said to be strictly ergodic if the homeomorphism has a unique, up to scaling, invariant Radon measure and every orbit of it is dense in the metric space.

SUMMARY: We study a perturbation of graph iterated function systems (graph IFS). In this study, we consider a family of graph IFSs with small parameter \(\epsilon >0\) such that some functions which compose this IFS converge to constant values as \(\epsilon \to 0\). Therefore, the limit graph IFS (unperturbed system) may have several Gibbs measures associated with the dimension of the limit set even if the parametrized graph IFS (perturbed system) possess a unique Gibbs measure \(\mu (\epsilon ,\cdot )\) for each \(\epsilon >0\). In this situation, we give a necessary and sufficient condition for convergence of \(\mu (\epsilon ,\cdot )\) in the case when the limit graph IFS has \(2\) or \(3\) Gibbs measures. In particular, this condition is composed of Perron eigenvalues of Ruelle operators.

SUMMARY: In this talk we deal with two types of random point configurations, spherical ensemble and the jittered sampling on the sphere. The former is a well-studied determinantal point process on the sphere, and the latter is one of the famous random sampling method. We compare these random point configurations with spherical designs, which are one of the non-random ^^ ^^ good” point configurations on the sphere in the viewpoint of \(p\)-frame potential. We also discuss other random point configurations on the sphere if possible.

SUMMARY: We consider the Schrödinger operator whose potential consists of the white noise multiplied by a decaying factor. This model can be defined as a symmetric operator for any realization of white noise, and it is almost surely self-adjoint under some mild conditions. The nature of the positive part of this operator is similar to the model considered by Kotani and Ushiroya (CMP vol. 115 (1988)).

SUMMARY: Limiting behaviors of one-dimensional diffusion processes with random potentials are studied. The potentials consist of specially contracted self-similar processes. The minimum processes and the maximum processes of the processes are also investigated.

SUMMARY: We discuss whether a random function (or a stochastic differential as an extension) is identified and how it is reconstructed from the stochastic Fourier coefficients (SFCs). In this talk, we focus on Skorokhod type SFC and give simple reconstruction formulas using Wiener chaos decomposition from SFCs of Wiener functionals on a space with Haar measure.

SUMMARY: In this talk, we study quasiconvex programming with a reverse quasiconvex constraint. We introduce affine and quasiaffine characterizations of a reverse quasiconvex constraint. By using these characterizations, we show necessary optimality conditions for the problem in terms of Greenberg–Pierskalla subdifferential. Additionally, we investigate surrogate duality for quasiconvex programming with a reverse quasiconvex constraint.

SUMMARY: In this talk, we consider an adaptive control approach in Markov decision process in order to solve a problem of multivariate Bayesian control chart. We show that there exist an average optimal adaptive and asymptotically discounted optimal policies.

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

SUMMARY: A distribution whose normalization constant is an A-hypergeometric polynomial is called an A-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. For sampling from an A-hypergeometric distribution, the first choice may be use of Markov chain Monte Carlo (MCMC) with moves generated by a Markov basis. In this talk, as an alternative to MCMC methods, a direct sampling algorithm for general A-hypergeometric distribution will be presented. As an application of the exact sampler, sampling from random Young tableaux will be discussed. The Ferguson’s Dirichlet process is an example of such random Young tableaux. A popular direct sampler, such as the Blackwell–MacQueen’s urn scheme, does not work for random Young tableaux without infinite exchangeability. In contrast to the urn schemes, our direct sampler still works without exchangeability.

SUMMARY: Let \({\rm Spin}(2\ell +1)\) denote the spin group, represented as a subgroup of \({\rm SU}(2^{\ell })\) (spin representation). Let \(G\) be \({\rm SU}(\ell )\) or \({\rm Spin}(2\ell +1)\), and \(V\) be \(\mathbb {C}^{\ell }\) or \(\mathbb {C}^{2^{\ell }}\), respectively. Fix \(T>0\) and \(S^{1}\) denote the loop, viewed as the interval \([0,2T]\subset \mathbb {R}\) where the endpoints identified. Let \(H\) be a self-adjoint operator on \(V\). For any operator \(A\) on \(V\), let \(A(t):=e^{-itH}Ae^{itH},\ t\in \mathbb {R}.\) We show some formulas which give the value of \[ {\rm Tr} A_{n}(t_{n})\cdots A_{0}(t_{0})B_{1}(t_{1}')\cdots B_{m}(t_{m}'),\ 0=t_{0}<\cdots <t_{n}=T,\ 0<t_{1}'<\cdots <t_{m}'<T, \] which can be interpreted as a quantum expectation value, by a limit of probability measures on \(C^{\infty }\left (S^{1},G\right )\).

SUMMARY: We are concerned with the regularity of centered Gaussian processes \((Z_x( \omega ))_{x \in M}\) indexed by compact metric spaces \((M, \rho )\). We are to show as our main result that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance \(K(x,y) = E(Z_x Z_y)\) under the assumption that (i) there is an underlying Dirichlet structure on \(M\) which determines the Besov space regularity, and (ii) the operator \(K\) with kernel \(K(x, y)\) and the underlying operator \(A\) of the Dirichlet structure commute.
As an application of this result we investigate the case of compact homogeneous spaces and, in particular, the case where \(M\) is the sphere.

SUMMARY: Diffusion processes in a tube are direct product diffusion processes \(\mathbb Y\) of one dimensional diffusion processes \(X^{(1)}\) and skew product diffusions \(\Xi ,\) or the time changed process \(\mathbb X\) which is based on a positive continuous additive functional \(\Phi (t)\). The skew product \(\Xi \) are given by one dimensional diffusion processes \(\rm R\) and a spherical Brownian motion \(\Theta \) by means of positive continuous additive functional \({\bf f}(t)\). We show a limit theorem for a sequence of time changed process \({\mathbb X}_n\) under some assumptions for \({\rm R}_n\), \(\nu _n\) ( Revuz measure of \({\bf f}_n(t)\)), and underlying measure.

SUMMARY: We consider the third-order linear model for \(3^{m}\) factorials. In previous talk (MSJ Autumn Meeting 2017), we gave the A\(^{\ast }\)-optimal \(3^{m}\)-BTO designs of resolution \(\mathrm {R}^{\ast }(\{10,01\})\) derived from SA’s with the number of assemblies (\(=N\)) is less than the number of non-negligible factorial effects (\(=\nu (m)\)) and \(m\ge 6.\)

Let \(T\) be a \(3^{m}\)-BTO design of resolution \(\mathrm {R}^{\ast }(\{10,01\})\) derived from an SA with \(N\) assemblies, and further let \(\sigma ^{2}S_{T}(\alpha )\) (\(\alpha =0,1,2\)) be the trace of the variance-covariance matrix of the estimators based on \(T\). If \(S_{T}(\alpha )\le S_{T^{\ast }}(\alpha )\) for any \(T^{\ast },\) then \(T\) is said to be \(\mathrm {GA}_{\alpha }^{\ast }\)-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 present \(\mathrm {GA}_{\alpha }^{\ast }\)-optima \(3^{m}\)-BTO designs of resolution \(\mathrm {R}^{\ast }(\{10,01\})\) derived from SA’s for \(m=6,7,8,\) where \(N<\nu (m).\)

SUMMARY: In this talk, we propose a new measure for square contingency tables. We construct the measure on the basis of the weighted harmonic mean of the diversity index. We derive properties of the measure and introduce a new model, local symmetry model.

SUMMARY: We consider a test of complete independence in multi-dimensional contingency tables. We derive an expression for approximation of the null distribution of the test statistic based on asymptotic expansion. By using the continuous term of the expansion, we consider transformed statistics that increase the speed of convergence to a chi-square limiting distribution.

SUMMARY: For square contingency tables with ordered categories, there may be some cases that one wants to analyze them by considering collapsed tables with some adjacent categories combined in the original table. This presentation considers the point-symmetry model for collapsed square contingency tables and proposes a measure to represent the degree of departure from point-symmetry. Also this presentation gives approximate confidence interval for proposed measure.

SUMMARY: For the analysis of square contingency table, Caussinus (1965) pointed out that the symmetry model holds if and only if both the quasi-symmetry model and the marginal homogeneity model hold. This presentation proposes the covariance symmetry model and the decomposition theorem of the symmetry model into the covariance symmetry model and the marginal homogeneity model, which are different from Causinnus’s.

SUMMARY: This paper considers to estimate conditional mutual information given three sequences each of which is either continuous of discrete. The estimation generates a sequence of quantizations, estimate conditional mutual information of quantized values, and choose the maximum estimation value. It estimates continuous and discrete variables alike in a seemless manner. In particular, we prove two important properties. First, with probability one as the sample size goes to infinity, the obtained estimation is zero if and only if they are conditionally independent. Secondly, the estimation asymptotically converges to the true value. The procedure has been implemented in the CRAN package BNSL developed by J. Suzuki and J. Kawahara.

SUMMARY: In this talk, we consider estimation of eigenvalues in high-dimensional settings. First, we show that the sample eigenvalue is not a consistent estimator of the true eigenvalue for high-dimensional settings. Yata and Aoshima (2012, JMVA) proposed a new PCA method called the noise reduction (NR) methodology. The estimation of the eigenvalue by the NR method has a first-order consistency. We investigate more deeply the asymptotic behavior of the NR method. We give a new eigenvalue estimation and show that it holds the second-order consistency.

SUMMARY: In this talk, we consider the equality tests of covariance matrices for high-dimensional data. Aoshima and Yata (2017) proposed two eigenvalue models for high-dimensional data and constructed two-sample test procedures. One is called strongly spiked eigenvalue (SSE) model and the other one is called non-SSE (NSSE) model. Ishii et al. (2016) proposed an equality test of two covariance matrices under the SSE model. Li and Chen (2012) proposed a test procedure under the NSSE model. We evaluate the test statistic of Li and Chen (2012) under the SSE model and give new test procedures by using the noise-reduction method given by Yata and Aoshima (2012). We also compare our new test procedures with that given by Ishii et al. (2016).

SUMMARY: In this paper we discuss smoothed rank statistics for testing a location shift parameter of the two-sample problem. They are based on the discrete test statistics —the median and Wilcoxon’s rank sum tests. For the one-sample problem, Maesono et al. (2017) reported that some nonparametric discrete tests have a problem with their \(p\)-values because of their discreteness. The \(p\)-values of the Wilcoxon’s test are frequently smaller than those of the median test in tail area. This causes an arbitrary choice of the median and Wilcoxon’s rank sum tests. In order to conquer this problem, we propose smoothed versions of those tests. The smoothed tests inherit good properties of the original tests, and asymptotically equivalent to the original test statistics. We study significance probabilities and local asymptotic powers of the proposed tests.

SUMMARY: Inference for volatility under finite time horizon becomes non-ergodic statistics. The quasi-maximum likelihood estimator and the Bayesian type estimator of the volatility parameter are asymptotically mixed normal in general. Asymptotic expansion in non-ergodic systems is then indispensable to develop the higher-order inferential theory for volatility. We present asymptotic expansion of a martingale having a mixed normal limit. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. We discuss its application to the power variation of a diffusion process. Identification of the random symbols is an issue.

SUMMARY: Asymptotic expansion of the distribution of the Skorohod integral jointly with a reference variable is derived. We introduce a second-order interpolation formula in frequency domain to expand a characteristic functional and combine it with the scheme developed in the martingale expansion. Random symbols are used for expressing the asymptotic expansion formula. Quasi tangent, quasi torsion and modified quasi torsion are introduced in this paper. This is a joint work with D. Nualart.

SUMMARY: In this presentation, we develop the estimation theory for Whittle functional of high-dimensional non-Gaussian dependent processes. Using a sample version based on a thresholded periodogram matrix, we introduce a thresholded Whittle estimator of unknown parameter, and elucidate its asymptotics. It is shown that the thresholded Whittle estimator is a \(\sqrt {n}\)-consistent estimator of the unknown parameter, and that the standardized version has the asymptotic normality. Some numerical studies illuminate an interesting feature of the results. Concretely, for high-dimensional AR(2), we compared the difference of RMSE between the usual Whittle estimator \(\hat {\theta }_{w}\) and the thresholded estimator \(\hat {\theta }_{w,th}\), leading to a conclusion that \(\hat {\theta }_{w,th}\) is better than \(\hat {\theta }_{w}\).

SUMMARY: Consider a linear regression model: \(Y_t=z'_t\beta +\varepsilon _t\) where \(\{\varepsilon _t\}\) is a stationary process with mean zero and spectral density \(f(\lambda )\), and \(z_t\) is a known nonrandom function vector of \(t\). In this talk, it is desired to discuss the LASSO estimator of \(\beta \) when \(\{\varepsilon _t\}\) is a long-memory strictly stationary process (i.e. \(f(\lambda )\) is unbounded at the origin) all of whose moments exist and has the infinite moving average representation, and when the dimension of \(\beta \) defined as \(p\) increases with sample size \(n\). An interesting property of the LASSO estimators is shown.

SUMMARY: This talk deals with the proportional hazards model proposed by D. R. Cox in a high-dimensional and sparse setting for a regression parameter. To estimate the regression parameter, the Dantzig selector is applied. The variable selection consistency of the Dantzig selector for the model will be proved. This property enables us to reduce the dimension of the parameter and to construct asymptotically normal estimators for the regression parameter and the cumulative baseline hazard function.

SUMMARY: Recently, we often observe the heavy-tailed time series data in variety of fields, and it is unfeasible to apply the classical likelihood ratio-based method to such data directly. To overcome the difficulty, this talk constructs the self-weighted generalized empirical likelihood (SW-GEL) statistic for possibly infinite variance processes, and elucidates the local asymptotic power of the SW-GEL statistic. The self-weighting method proposed by Ling (2005, JRSS) enables us to control effects brought by the infinite variance of underlying time series models. By the self-weighting method, the proposed statistic converges to the non-central chi-square distribution under the local alternatives. This talk also introduces the selection procedure of tuning parameters in self-weights based on the local asymptotic power.

SUMMARY: It is well known that the property of local asymptotic normality (LAN) allows us to discuss the asymptotic efficiency of estimation via minimax theorems. We proved LAN property for symmetric stable processes and one-sided stable processes under high-frequency observations using non-diagonal rate matrices depending on the parameter to be estimated. In contrast to the classical LAN families in the literature, non-diagonal rate matrices are inevitable.