Yuji Kasahara

Weighted sums of I.I.D. Random variables attracted to integrals of stable processes

SummaryUnder the assumption of the convergence of partial sum processesZn(t) of i.i.d. random variables to an α-stable Lévy processZ(σ)(t), 0<α≦2, the convergence of weighted sums ∫fn(u)dZn(u) to ∫f(u)dZ(α)(u) is studied. The general convergence result is then applied to examine the domain of attraction of the fractional stable process.

Log-fractional stable processes

The first problem attacked in this paper is answering the question whether all 1/[alpha]-self-similar [alpha]-stable processes with stationary increments are [alpha]-stable motions. The answer is yes for [alpha] = 2, no for 1[less-than-or-equals, slant][alpha]

Functional limit theorems for weighted sums of I.I.D. random variables

On etudie des theoremes limites fonctionnels pour des sommes ponderees, specialement pour des poids concernant les methodes de sommabilite classiques telles que celles de Cesaro, Abel, Borel et Euler

On a generalized arc-sine law for one-dimensional diffusion processes

Laws of the occupation times on a half line are studied for one-dimensional diffusion processes. The asymptotic behavior of the distribution function is determined in terms of the speed measure.

Stability theorem for stochastic differential equations with jumps

Convergence in law of solutions of SDE having jumps is discussed assuming suitable convergence of the coefficients under a situation where the point process approaches a Poisson point process. As an application the asymptotic behavior of certain stochastic processes such as storage processes and random walks is also discussed.

A functional limit theorem for trimmed sums

This paper proves a functional limit theorem for Stiglers result on the heavily trimmed sums of i.i.d. random variables. The limiting process will be expressed as a functional of a Kiefer process and we shall also see that it is a Brownian motion if and only if asymptotic normality holds.

Occupation time theorems for a class of one-dimensional diffusion processes

SummaryThe long time asymptotic behavior of the occupation times on a half line is studied for a class of one-dimensional diffusion processes whose excursion intervals have very heavy tail probability

Remarks on Tauberian theorem of exponential type and Fenchel-Legendre transform

Let ( ), ≥ 0 be a nondecreasing right-continuous function such that (0) = 0. The asymptotics of and its Laplace-Stieltjes transform ω( ) = ∫∞ 0 − ( ) are closely linked and results in which we pass from ( ) to ω( ) are called Abelian theorems and ones in converse direction are called Tauberian, and they play a very important role in probability theory. A most well-known result on this subject is Karamata’s theorem (cf. Chapter 1 of [1]). Also the cases when ω( ) and ( ) vary exponentially are treated by many authors (e.g. [2], [3], [4], [8], [9]. See also Chapter 4 of [1]). Among them [2] studied the relationship between the limit of (1/λ) log (1/φ(λ)) as λ → ∞ and that of the Laplace-Stieltjes transform modified as

Limit theorems for trimmed sums

This paper studies the heavily trimmed sums (*) ∑[ns] + 1[nt]Xj(n), where {Xj(n)}j = 1n are the order statistics from independent random variables {X1,...,Xn} having a common distributionF. The main theorem gives the limiting process of (*) as a process oft. More smoothly trimmed sums like ∑j = 1[nt]J(j/n)Xj(n) are also discussed.

Asymptotic Behavior of the Transition Density of an Ergodic Linear Diffusion

Positive recurrent diffusions on the line are treated. We study the asymptotic behavior of the transition density in the long term. The problem is equivalent to the study of Krein’s correspondence for bounded strings. 2010 Mathematics Subject Classification: Primary 60J35; Secondary 60J60, 34B24.