arXiv subjects

Tetsuya Hattori

Publications and source records attributed to Tetsuya Hattori.

16 recordsLinked to original sources

arXiv2025

Choquet integration with submodular function on measurable space with sigma-algebra generating chain

Based on a study of a formula representing submodular set function as a supremum of measures dominated by the set function, we present a corresponding formula for a Choquet integration with respect to the set function, on a measurable space which has a chain of measurable set generating the sigma-algebra. As an application we reproduce a basic formula in mathematical finance on law invariant coherent risk measures. We also study a recursion relation of set functions for which the representation formula characterizes the fixed point.

math.PR↗

arXiv2024

An elementary proof of representation of submodular function as an supremum of measures on $σ$-algebra with totally ordered generating class

We give an alternative proof of a fact that a finite continuous non-decreasing submodular set function on a measurable space can be expressed as a supremum of measures dominated by the function, if there exists a class of sets which is totally ordered with respect to inclusion and generates the sigma-algebra of the space. The proof is elementary in the sense that the measure attaining the supremum in the claim is constructed by a standard extension theorem of measures. As a consequence, a uniquness of the supremum attaining measure also follows. A Polish space is an examples of the measurable space which has a class of totally ordered sets that generates the Borel sigma-algebra.

math.FA↗

arXiv2016

Cancellation of fluctuation in stochastic ranking process with space-time dependent intensities,

We consider the stochastic ranking process with space-time dependent unbounded jump rates for the particles. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution in the infinite particle limit. We assume topology of weak convergence for the space of distributions, which implies that the fluctuations among particles with different jump rates cancel in the limit. The results are proved by first finding an auxiliary stochastic ranking process, for which a strong law of large numbers is applied, and then applying a multi time recursive Gronwall's inequality. The limit has a representation in terms of non-Markovian processes which we call point processes with last-arrival-time dependent intensities.

math.PR↗

arXiv2016

Doubly uniform complete law of large numbers for independent point processes

We prove a law of large numbers in terms of complete convergence of independent random variables taking values in increments of monotone functions, with convergence uniform both in the initial and the final time. The result holds also for the random variables taking values in functions of $2$ parameters which share similar monotonicity properties as the increments of monotone functions. The assumptions for the main result are the Hölder continuity on the expectations as well as moment conditions, while the sample functions may contain jumps. In particular, we can apply the results to point processes (counting processes) which lack Markov or martingale type properties.

math.PR↗

arXiv2014

Point process with last-arrival-time dependent intensity and 1-dimensional incompressible fluid system with evaporation

We consider an infinite system of quasilinear first-order partial differential equations, generalized to contain spacial integration, which describes an incompressible fluid mixture of infinite components in a line segment whose motion is driven by unbounded and space-time dependent evaporation rates. We prove unique existence of the solution to the initial-boundary value problem, with conservation-of-fluid condition at the boundary. The proof uses a map on the space of collection of characteristics, and a representation based on a non-Markovian point process with last-arrival-time dependent intensity.

math.AP↗

arXiv2012

Stochastic ranking process with space-time dependent intensities

We consider the stochastic ranking process with space-time dependent jump rates for the particles. The process is a simplified model of the time evolution of the rankings such as sales ranks at online bookstores. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution, and also the tagged particle processes converge almost surely, in the infinite particle limit. The limit distribution is characterized by a system of inviscid Burgers-like integral-partial differential equations with evaporation terms, and the limit process of a tagged particle is a motion along a characteristic curve of the differential equations except at its Poisson times of jumps to the origin.

math.PR↗

arXiv2010

Stochastic ranking process with time dependent intensities

We consider the stochastic ranking process with the jump times of the particles determined by Poisson random measures. We prove that the joint empirical distribution of scaled position and intensity measure converges almost surely in the infinite particle limit. We give an explicit formula for the limit distribution and show that the limit distribution function is a unique global classical solution to an initial value problem for a system of a first order non-linear partial differential equations with time dependent coefficients.

math.PR↗

arXiv2009

Hydrodynamic limit of move-to-front rules and search cost probabilities

We study a hydrodynamic limit approach to move-to-front rules, namely, a scaling limit as the number of items tends to infinity, of the joint distribution of jump rate and position of items. As an application of the limit formula, we present asymptotic formulas on search cost probability distributions, applicable for general jump rate distributions.

math.PR↗

arXiv2009

Existence of an infinite particle limit of stochastic ranking process

We study a stochastic particle system which models the time evolution of the ranking of books by online bookstores (e.g., Amazon). In this system, particles are lined in a queue. Each particle jumps at random jump times to the top of the queue, and otherwise stays in the queue, being pushed toward the tail every time another particle jumps to the top. In an infinite particle limit, the random motion of each particle between its jumps converges to a deterministic trajectory. (This trajectory is actually observed in the ranking data on web sites.) We prove that the (random) empirical distribution of this particle system converges to a deterministic space-time dependent distribution. A core of the proof is the law of large numbers for {\it dependent} random variables.

math.PR↗

arXiv2008

Equation of motion for incompressible mixed fluid driven by evaporation and its application to online rankings

We give a unique classical solution to initial value problem for a system of partial differential equations for the densities of components of one dimensional incompressible fluid mixture driven by evaporation. Motivated by the known fact that the solution appears as an infinite particle limit of stochastic ranking processes, which is a simple stochastic model of time evolutions of e.g., Amazon Sales Ranks, we collected data from the web and performed statistical fits to our formula. The results suggest that the fluid equations and solutions may have an application in the analysis of online rankings.

math-ph↗

arXiv2008

Mathematical analysis of long tail economy using stochastic ranking processes

We present a new method of estimating the distribution of sales rates of, e.g., book titles at an online bookstore, from the time evolution of ranking data found at websites of the store. The method is based on new mathematical results on an infinite particle limit of the stochastic ranking process, and is suitable for quantitative studies of the long tail structure of online retails. We give an example of a fit to the actual data obtained from Amazon.co.jp, which gives the Pareto slope parameter of the distribution of sales rates of the book titles in the store.

q-fin.GN↗

arXiv2006

Uniqueness of fixed point of a two-dimensional map obtained as a generalization of the renormalization group map associated to the self-avoiding paths on gaskets

Let $W(x,y) = a x^3 + b x^4 + f_5 x^5 + f_6 x^6 + (3 a x^2)^2 y + g_5 x^5 y + h_3 x^3 y^2 + h_4 x^4 y^2 + n_3 x^3 y^3 + a_{24} x^2 y^4 + a_{05} y^5 + a_{15} x y^5 + a_{06} y^6$, and $X=\frac{\partial W}{\partial x}$, $Y=\frac{\partial W}{\partial y}$, where the coefficients are non-negative constants, with $a>0$, such that $X^{2}(x,x^{2})-Y(x,x^{2})$ is a polynomial of $x$ with non-negative coefficients. Examples of the 2 dimensional map $Φ: (x,y)\mapsto (X(x,y),Y(x,y))$ satisfying the conditions are the renormalization group (RG) map (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point $(x_f,y_f)$ of $Φ$ in the invariant set $\{(x,y)\in R^2\mid x^2\ge y\}\setminus\{0\}$.

math-ph↗

arXiv1997

Mathematical Derivation of Chiral Anomaly in Lattice Gauge Theory with Wilson's Action

Chiral U(1) anomaly is derived with mathematical rigor for a Euclidean fermion coupled to a smooth external U(1) gauge field on an even dimensional torus as a continuum limit of lattice regularized fermion field theory with the Wilson term in the action. The present work rigorously proves for the first time that the Wilson term correctly reproduces the chiral anomaly.

hep-lat↗

arXiv1995

Restoration of isotropy on fractals

We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions, random walks, resistor networks, and Gaussian field theories. The mechanism is unique in that it is absent in spaces with translational invariance, while universal in that it is observed in a wide class of fractals.

cond-mat↗

arXiv1994

Transition density of diffusion on Sierpinski gasket and extension of Flory's formula

Some problems related to the transition density u(t,x) of the diffusion on the Sierpinski gasket are considerd, based on recent rigorous results and detailed numerical calculations. The main contents are an extension of Flory's formula for the end-to-end distance exponent of self-avoiding walks on the fractal spaces, and an evidence of the oscillatory behavior of u(t,x) on the Sierpinski gasket.

cond-mat↗

arXiv1992

Improvement of efficiency in generating random $U(1)$ variables with Boltzmann distribution

A method for generating random $U(1)$ variables with Boltzmann distribution is presented. It is based on the rejection method with transformation of variables. High efficiency is achieved for all range of temparatures or coupling parameters, which makes the present method especially suitable for parallel and pipeline vector processing machines. Results of computer runs are presented to illustrate the efficiency. An idea to find such algorithms is also presented, which may be applicable to other distributions of interest in Monte Carlo simulations.

hep-lat↗