arXiv subjects

Ryuya Namba

Publications and source records attributed to Ryuya Namba.

16 recordsLinked to original sources

arXiv2025

Asymptotic behaviors of fractional binomial distributions derived from the generalized binomial theorem

A fractional binomial distribution, introduced by Hino and Namba (2024) via the generalized binomial theorem, is a fractional variant of the classical binomial distribution. Building upon previous work that established limit theorems, such as the weak law of large numbers and the central limit theorem, for fractional binomial distributions, this paper further investigates their asymptotic behaviors. Specifically, we derive the large and moderate deviation principles and a Berry--Esseen type estimate for these distributions.

math.PR↗

arXiv2025

On some properties of a positive linear operator via the Moran model in population genetics

We introduce a positive linear operator acting on the Banach space of all continuous functions on the unit interval via the Moran model studied in population genetics. We show that this operator, named the Moran operator, uniformly approximates every continuous function on the unit interval. Furthermore, some limit theorems for the iterates of the Moran operator are obtained.

math.PR↗

arXiv2024

Volatility estimation from a view point of entropy

In the present paper, we first revisit the volatility estimation approach proposed by N. Kunitomo and S. Sato, and second, we show that the volatility estimator proposed by P. Malliavin and M.E. Mancino can be understood in a unified way by the approach. Third, we introduce an alternative estimator that might overcome the inconsistency caused by the microstructure noise of the initial observation.

math.ST↗

arXiv2024

Fractional binomial distributions induced by the generalized binomial theorem and their applications

We develop a fractional extension of the classical binomial distribution and the associated Bernstein operator, formulated within the framework of the generalized binomial theorem (Hara and Hino [Bull.\ London Math.\ Soc. \textbf{42} (2010), 467--477]). This provides a new probabilistic structure not representable as the law of the sum of independent and identically distributed random variables. Despite this nonstandard nature, we establish several of its fundamental analytic and probabilistic properties, including limit theorems,through a unified framework based on the generalized binomial theorem.We further analyze the properties of the fractional Bernstein operator associated with the fractional binomial distribution. In particular, we prove that the iterates of the operator converge to a generalized Wright--Fisher diffusion semigroup after a proper diffusive rescaling.

math.PR↗

arXiv2023

Iterates of multidimensional Bernstein-type operators and diffusion processes in population genetics

The Bernstein operator is known as a typical example of positive linear operators which uniformly approximates continuous functions on $[0, 1]$. In the present paper, we introduce a multidimensional extension of the Bernstein operator which is associated with a transition probability of a certain discrete Markov chain. In particular, we show that the iterate of the multidimensional Bernstein-type operator uniformly converges to the Feller semigroup corresponding to the multidimensional Wright-Fisher diffusion process with mutation arising in the study of population genetics, together with its rate of convergence. The convergence of process-level is obtained as well. Moreover, by taking the limit as both the number of iterate and the dimension of the Bernstein-type operator tend to infinity simultaneously, we prove that the iterate of the multidimensional Bernstein-type operator uniformly converges to the Feller semigroup corresponding to a probability measure-valued Fleming-Viot process with mutation.

math.PR↗

arXiv2023

The SIML method without microstructure noise

The SIML (abbreviation of Separating Information Maximal Likelihood) method, has been introduced by N. Kunitomo and S. Sato and their collaborators to estimate the integrated volatility of high-frequency data that is assumed to be an It\^o process but with so-called microstructure noise. The SIML estimator turned out to share many properties with the estimator introduced by P. Malliavin and M.E. Mancino. The present paper establishes the consistency and the asymptotic normality under a general sampling scheme but without microstructure noise. Specifically, a fast convergence shown for Malliavin--Mancino estimator by E. Clement and A. Gloter is also established for the SIML estimator.

math.ST↗

arXiv2022

Edgeworth expansions for centered random walks on covering graphs of polynomial volume growth

Edgeworth expansions for random walks on covering graphs with groups of polynomial volume growths are obtained under a few natural assumptions. The coefficients appearing in this expansion depends on not only geometric features of the underlying graphs but also the modified harmonic embedding of the graph into a certain nilpotent Lie group. Moreover, we apply the rate of convergence in Trotter's approximation theorem to establish the Berry-Esseen type bound for the random walks.

math.PR↗

arXiv2022

Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth

Moderate deviation principles (MDPs) for random walks on covering graphs with groups of polynomial volume growth are discussed in a geometric point of view. They deal with any intermediate spatial scalings between those of laws of large numbers and those of central limit theorems. The corresponding rate functions are given by quadratic forms determined by the Albanese metric associated with the given random walks. We apply MDPs to establish laws of the iterated logarithm on the covering graphs by characterizing the set of all limit points of the normalized random walks.

math.PR↗

arXiv2022

Rate of convergence in Trotter's approximation theorem and its applications

The celebrated Trotter approximation theorem provides a sufficient condition for the convergence of a sequence of operator semigroups in terms of the corresponding sequence of infinitesimal generators. There exist a few results on the rate of convergence in Trotter's theorem under some constraints. In the present paper, a new rate of convergence in Trotter's theorem in full generality is given. Moreover, we see that this rate of convergence works well to obtain quantitative estimates for some limit theorems in probability theory.

math.FA↗

arXiv2022

Limit theorems for iterates of the Szász-Mirakyan operator in probabilistic view

The Szász-Mirakyan operator is known as a positive linear operator which uniformly approximates a certain class of continuous functions on the half line. The purpose of the present paper is to find out limiting behaviors of the iterates of the Szász-Mirakyan operator in a probabilistic point of view. We show that the iterates of the Szász-Mirakyan operator uniformly converges to a continuous semigroup generated by a second order degenerate differential operator. A probabilistic interpretation of the convergence in terms of a discrete Markov chain constructed from the iterates and a limiting diffusion process on the half line is captured as well.

math.PR↗

arXiv2022

Random walks on crystal lattices and multiple zeta functions

Crystal lattices are known to be one of the generalizations of classical periodic lattices which can be embedded into some Euclidean spaces properly. As to make a wide range of multidimensional discrete distributions on Euclidean spaces more treatable, multidimensional Euler products and multidimensional Shintani zeta functions on crystal lattices are introduced. They are completely different from existing Ihara zeta functions on graphs in that our zeta functions are defined on crystal lattices directly. Via a concept of periodic realizations of crystal lattices, we make it possible to provide many kinds of multidimensional discrete distributions explicitly. In particular, random walks on crystal lattices whose range is infinite and such random walks whose range is finite are constructed by multidimensional Euler products and multidimensional Shintani zeta functions, respectively. We give some comprehensible examples as well.

math.PR↗

arXiv2022

Asymptotic behaviors of convolution powers of the Riemann zeta distribution

In probability theory, there exist discrete and continuous distributions. Generally speaking, we do not have sufficient kinds and properties of discrete ones compared to the continuous ones. In this paper, we treat the Riemann zeta distribution as a representative of few known discrete distributions with infinite supports. Some asymptotic behaviors of convolution powers of the Riemann zeta distribution are discussed.

math.PR↗

arXiv2022

Monotonic normalized heat diffusion for regular bipartite graphs with four eigenvalues

Let $X=(V, E)$ be a finite regular graph and $H_t(u, v), \, u, v \in V$, the heat kernel on $X$. We prove that, if the graph $X$ is bipartite and has four distinct Laplacian eigenvalues, the ratio $H_t(u, v)/H_t(u, u), \, u, v \in V,$ is monotonically non-decreasing as a function of $t$. The key to the proof is the fact that such a graph is an incidence graph of a symmetric 2-design.

math.CO↗

arXiv2019

Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part I

In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the nilpotent Lie group equipped with the Albanese metric. The drift term arises from the non-symmetry of the random walk and it vanishes when the random walk is symmetric. Furthermore, by imposing the "centered condition", we establish a functional CLT (i.e., Donsker-type invariance principle) in a Hoelder space over the nilpotent Lie group. The functional CLT is extended to the case where the realization is not necessarily harmonic. We also obtain an explicit representation of the limiting diffusion process on the nilpotent Lie group and discuss a relation with rough path theory. Finally, we give several examples of random walks on nilpotent covering graphs with explicit computations.

math.PR↗

arXiv2019

Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part II

In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We first prove a semigroup CLT for the family of random walks by realizing the nilpotent covering graph into a nilpotent Lie group via discrete harmonic maps. The limiting diffusion semigroup is generated by the homogenized sub-Laplacian with a constant drift of the asymptotic direction on the nilpotent Lie group, which is equipped with the Albanese metric associated with the symmetrized random walk. We next prove a functional CLT (i.e., Donsker-type invariance principle) in a Holder space over the nilpotent Lie group by combining the semigroup CLT, standard martingale techniques, and a novel pathwise argument inspired by rough path theory. Applying the corrector method, we finally extend these CLTs to the case where the realizations are not necessarily harmonic.

math.PR↗

arXiv2017

A remark on a central limit theorem for non-symmetric random walks on crystal lattices

Recently, Ishiwata, Kawabi and Kotani [2] proved two kinds of central limit theorems for non-symmetric random walks on crystal lattices from the view point of discrete geometric analysis. In the present paper, we obtain yet another kind of the central limit theorem for them. Our argument is based on a measure-change technique due to Alexopoulos [1].

math.PR↗