arXiv subjects

Muneya Matsui

Publications and source records attributed to Muneya Matsui.

At least 19 recordsLinked to original sources

arXiv2026

Can we further improve the central limit theorem for stationary $\rho$-mixing sequences $?$

Through the seminal works of Peligrad and Bradley, the existing sufficient conditions for the central limit theorem for $\rho$-mixing sequences are known to be very close to necessary. Nevertheless, we show that further improvements to these conditions are achievable when the $\rho$-mixing coefficients are non-summable, bridging a long-standing gap in the borderline case. The key idea is an iterative block-size technique that enables precise evaluations of the slowly varying variance components. Finally, known examples and new extensions--including iterated logarithmic mixing rates--are investigated to demonstrate the sharpness of our criteria, revealing a clear parameter structure and posing related open problems.

math.PR↗

arXiv2026

Moments for self-normalized partial sums

We consider a regularly varying stationary sequence of random variables (Xt) with tail index ___ < 2. For these sequences we study the joint convergence of sums, `p- type moduli and maxima. We focus on ratio statistics, including the studentized sums and sums normalized by the corresponding maxima, and study the existence of moments for the limit ratios. We consider particular examples of processes (Xt) whose limit ratios possess all moments. But, in contrast to the latter situation, there also exist sequences (Xt) where certain moments of the limit ratio are in___nite. This phenomenon results from extremal clusters in the sequence.

math.PR↗

arXiv2026

Heavy Tails and Predictive Ability Testing

We study the asymptotic behaviour of widely used tests for evaluating and comparing predictive accuracy when forecast errors exhibit heavy tails. In particular, when loss differentials have infinite variance, the Diebold-Mariano test statistic converges to a nonstandard limit involving non-Gaussian stable random variables. As a consequence, conventional critical values can yield severely distorted inference: a nominal 5$\%$ test may reject a true null as often as 70$\%$ of the time. To establish these results, we develop a new stable limit theorem for strongly mixing, infinite-variance time series processes. Building on this theory, we consider sub-sampling-based inference that remains valid irrespective of tail-heaviness and requires no estimation of long-run variances or tail indices. An application to risk forecasts for emerging-market exchange rates shows that accounting for heavy tails can substantially alter conclusions about predictive performance relative to standard procedures.

stat.ME↗

arXiv2025

The Gaussian central limit theorem for a stationary time series with infinite variance

We consider a borderline case: the central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the iid case a well-known sufficient condition for this central limit theorem is regular variation of the marginal distribution with tail index $\alpha=2$. In the dependent case we assume the stronger condition of sequential regular variation of the time series with tail index $\alpha=2$. We assume that a sample of size $n$ from this time series can be split into $k_n$ blocks of size $r_n\to\infty$ such that $r_n/n\to 0$ as $n\to\infty$ and that the block sums are asymptotically independent. Then we apply classical central limit theory for row-wise iid triangular arrays. The necessary and sufficient conditions for such independent block sums will be verified by using large deviation results for the time series. We derive the central limit theorem for $m$-dependent sequences, linear processes, stochastic volatility processes and solutions to affine stochastic recurrence equations whose marginal distributions have infinite variance and are regularly varying with tail index $\alpha=2$.

math.PR↗

arXiv2023

On convolution closure properties of subexponentiality approaching from densities

Non-closedness of subexponentiality by the convolution operation is well-known. We go a step further and show that subexponentiality and non-subexponentiality are generally changeable by the convolution. We also give several conditions, by which (non-) subexponentiality is kept. Most results are given with densities, which are easily converted to those for distributions. As a by-product, we give counterexamples to several past results, which were used to derive the non-closedness of the convolution, and modify the original proof.

math.PR↗

arXiv2023

Mle-equivariance, data transformations and invariant tests of fit

We define data transformations that leave certain classes of distributions invariant, while acting in a specific manner upon the parameters of the said distributions. It is shown that under such transformations the maximum likelihood estimators behave in exactly the same way as the parameters being estimated. As a consequence goodness--of--fit tests based on standardized data obtained through the inverse of this invariant data--transformation reduce to the case of testing a standard member of the family with fixed parameter values. While presenting our results, we also provide a selective review of the subject of equivariant estimators always in connection to invariant goodness--of--fit tests. A small Monte Carlo study is presented for the special case of testing for the Weibull distribution, along with real--data illustrations.

math.ST↗

arXiv2023

Self-normalized partial sums of heavy-tailed time series

We study the joint limit behavior of sums, maxima and $\ell^p$-type moduli for samples taken from an $\mathbb{R}^d$-valued regularly varying stationary sequence with infinite variance. As a consequence, we can determine the distributional limits for ratios of sums and maxima, studentized sums, and other self-normalized quantities in terms of hybrid characteristic functions and Laplace transforms. These transforms enable one to calculate moments of the limits and to characterize the differences between the iid and stationary cases in terms of indices which describe effects of extremal clustering on functionals acting on the dependent sequence.

math.PR↗

arXiv2023

Asymptotics of densities of first passage times for spectrally negative Lévy processes

We study a first passage time of a Lévy process over a positive constant level. In the spectrally negative case we give conditions for absolutely continuity of the distributions of the first passage times. The tail asymptotics of their densities are also clarified, where the asymptotics depend on tail behaviour of the corresponding Lévy measures. We apply our results to the mathematical finance, in particular, the credit default swap pricing.

math.PR↗

arXiv2023

Local subexponentiality and infinitely divisible distributions

We completely characterize $Δ$- and local subexponentialities of positive-half compound Poisson distributions and extend the characterization on two-sided distributions. Moreover, $Δ$-subexponentiality of infinitely divisible distributions is characterized with new conditions, and local subexponentiality is newly characterized in the two-sided case. In the process closedness properties of these subexponentialities are derived, particularly for distributions on $\R$. Most results are obtained by exploiting monotonic-type assumptions. We apply our results to distributions of supremum of a random work and a randomly stopped iid sum.

math.PR↗

arXiv2023

Subexponentialiy of densities of infinitely divisible distributions

We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its Lévy measure and the tail equivalence between the density and its Lévy measure density, under monotonic-type assumptions on the Lévy measure density. The key assumption is that tail of the Lévy measure density is asymptotic to a non-increasing function or is eventually non-increasing. Our conditions are novel and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the subexponentiality of densities have been derived such as closure properties of [ convolution, convolution roots and asymptotic equivalence ] and the factorization property. Moreover, we illustrate that the results are applicable for developing the statistical inference of subexponential infinitely divisible distributions which are absolutely continuous.

math.PR↗

arXiv2022

Trigonometrically approximated maximum likelihood estimation for stable law

A trigonometrically approximated maximum likelihood estimation for $α$-stable laws is proposed. The estimator solves the approximated likelihood equation, which is obtained by projecting a true score function on the space spanned by trigonometric functions. The projected score is expressed only by real and imaginary parts of the characteristic function and their derivatives, so that we can explicitly construct the targeting estimating equation. We study the asymptotic properties of the proposed estimator and show consistency and asymptotic normality. Furthermore, as the number of trigonometric functions increases, the estimator converges to the exact maximum likelihood estimator, in the sense that they have the same asymptotic law. Simulation studies show that our estimator outperforms other moment-type estimators, and its standard deviation almost achieves the Cramér--Rao lower bound. We apply our method to the estimation problem for $α$-stable Ornstein--Uhlenbeck processes in a high-frequency setting. The obtained result demonstrates the theory of asymptotic mixed normality.

math.ST↗

arXiv2022

Distance covariance for random fields

We study an independence test based on distance correlation for random fields $(X,Y)$. We consider the situations when $(X,Y)$ is observed on a lattice with equidistant grid sizes and when $(X,Y)$ is observed at random locations. We provide asymptotic theory for the sample distance correlation in both situations and show bootstrap consistency. The latter fact allows one to build a test for independence of $X$ and $Y$ based on the considered discretizations of these fields. We illustrate the performance of the bootstrap test in a simulation study involving fractional Brownian and infinite variance stable fields. The independence test is applied to Japanese meteorological data, which are observed over the entire area of Japan.

math.ST↗

arXiv2022

Tails of bivariate stochastic recurrence equation with triangular matrices

We study bivariate stochastic recurrence equations with triangular matrix coefficients and we characterize the tail behavior of their stationary solutions ${\bf W} =(W_1,W_2)$. Recently it has been observed that $W_1,W_2$ may exhibit regularly varying tails with different indices, which is in contrast to well-known Kesten-type results. However, only partial results have been derived. Under typical "Kesten-Goldie" and "Grey" conditions, we completely characterize tail behavior of $W_1,W_2$. The tail asymptotics we obtain has not been observed in previous settings of stochastic recurrence equations.

math.PR↗

arXiv2020

Tail indices for AX+B recursion with triangular matrices

Multivariate stochastic recurrence equations (SREs) are investigated when coefficients are triangular matrices. If coefficient matrices of SREs have all strictly positive elements, the Kesten's classical result yields solutions with regularly varying tails such that the tail indices of solutions are the same through coordinates. This framework is too restrictive for applications. In order to widen the applicability of the SREs, we study SREs with triangular matrix coefficients and prove that they have regularly varying solutions which may exhibit coordinate-wisely different tail exponents. We also specify the coefficients for regularly varying tails. Several applications are suggested for GARCH models.

math.PR↗

arXiv2019

A characterization of ARMA and Fractional ARIMA models with infinitely divisible innovations

The object of this paper is to study the asymptotic dependence structure of the linear time series models with infinitely divisible innovations by the use of their characteristic functions. Autoregressive moving-average (ARMA) models and fractional autoregressive integrated moving-average (FARIMA) models are analyzed. As examples of infinitely divisible innovations, the class of radially absolute continuous distributions and general non-symmetric stable distributions are considered. The finite dimensional distributionsn of these models are also obtained.

math.ST↗

arXiv2019

Asymptotics of maximum likelihood estimation for stable law with continuous parameterization

Asymptotics of maximum likelihood estimation for $α$-stable law are analytically investigated with a continuous parameterization. The consistency and asymptotic normality are shown on the interior of the whole parameter space. Although these asymptotics have been provided with Zolotarev's $(B)$ parameterization, there are several gaps between. Especially in the latter, the density, so that scores and their derivatives are discontinuous at $α=1$ for $β\neq 0$ and usual asymptotics are impossible. This is considerable inconvenience for applications. By showing that these quantities are smooth in the continuous form, we fill gaps between and provide a convenient theory. We numerically approximate the Fisher information matrix around the Cauchy law $(α,β)=(1,0)$. The results exhibit continuity at $α=1,\,β\neq 0$ and this secures the accuracy of our calculations.

math.ST↗

arXiv2019

Characterization of the tail behavior of a class of BEKK processes: A stochastic recurrence equation approach

We provide new, mild conditions for strict stationarity and ergodicity of a class of BEKK processes. By exploiting that the processes can be represented as multivariate stochastic recurrence equations, we characterize the tail behavior of the associated stationary laws. Specifically, we show that the each component of the BEKK processes is regularly varying with some tail index. In general, the tail index differs along the components, which contrasts most of the existing literature on the tail behavior of multivariate GARCH processes.

math.ST↗

arXiv2018

Distance covariance for discretized stochastic processes

Given an iid sequence of pairs of stochastic processes on the unit interval we construct a measure of independence for the components of the pairs. We define distance covariance and distance correlation based on approximations of the component processes at finitely many discretization points. Assuming that the mesh of the discretization converges to zero as a suitable function of the sample size, we show that the sample distance covariance and correlation converge to limits which are zero if and only if the component processes are independent. To construct a test for independence of the discretized component processes we show consistency of the bootstrap for the corresponding sample distance covariance/correlation.

math.ST↗