Ken-iti Sato

Semi-Selfsimilar Processes

A notion of semi-selfsimilarity of Rd-valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: the existence of the exponent for semi-selfsimilar processes; characterization of semi-selfsimilar processes as scaling limits; relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions, and examples; construction of semi-selfsimilar processes with stationary increments; and extension of the Lamperti transformation. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes. A wide-sense semi-selfsimilarity is defined and shown to be reducible to semi-selfsimilarity.

Diffusion processes and a class of Markov chains related to population genetics

We investigate convergence of sequences of Markov chains induced by direct product branching processes, which are defined by Karlin and McGregor [7] with the intention of unified treatment of Markov chains in population genetics. The induced Markov chains that we deal with in this paper have d types (d>2) with equal fertility (that is, selection does not occur), and mutation and migration are allowed for. Let R~ be the (d— l)-dimensional Euclidean space and let K be the set of x=(x19 •••, #r f_1)eΛ r ~ such that ^>0, •••, d-l tf^^O, 1 — Σ ^/^O Under some conditions, we prove convergence of the

Subordination and self-decomposability

Two facts are established concerning subordination and self-decomposability. (1) Any subordinated process arising from a Brownian motion with drift and a self-decomposable subordinator is self-decomposable. (2) Self-decomposable distributions of type G are not necessarily of type GL. Consequences of the first fact on smoothness of the distributions are discussed.

Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes

Properties of the law μ of the integral ∫ ∞ 0 c -N t- dY t are studied, where c > 1 and {(N t , Y t ), t ≥ 0) is a bivariate Levy process such that {N t } and {Y t } are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalized Ornstein-Uhlenbeck process. The law μ is parametrized by c, q and r, where p = 1 - q - r, q, and r are the normalized Levy measure of {(N t , Y t )} at the points (1,0), (0, 1) and (1, 1), respectively. It is shown that, under the condition that p > 0 and q > 0, μ c,q,r is infinitely divisible if and only if r ≤ pq. The infinite divisibility of the symmetrization of μ is also characterized. The law μ is either continuous-singular or absolutely continuous, unless r = 1. It is shown that if c is in the set of Pisot-Vijayaraghavan numbers, which includes all integers bigger than 1, then μ is continuous-singular under the condition q > 0. On the other hand, for Lebesgue almost every c > 1, there are positive constants C 1 and C 2 such that 11 is absolutely continuous whenever q ≥ C 1 p ≥ C 2 r. For any c > 1 there is a positive constant C 3 such that μ is continuous-singular whenever q > 0 and max{q, r} ≤ C 3p . Here, if {N t } and {Y t } are independent, then r = 0 and q = b/(a + b).

Distributions of selfsimilar and semi-selfsimilar processes with independent increments

Relationships between marginal and joint distributions of selfsimilar processes with independent increments are shown in terms of the Urbanik-Sato-type nested subclasses of the class of selfdecomposable distributions. Similar results are also shown for semi-selfsimilar processes with independent increments.

Convergence to a diffusion of a multi-allelic model in population genetics

We consider a Markov chain on the d -dimensional ( d -alleli) non-negative lattice points with the sum of components being N, for which one-step transition consists of two stages—independent reproduction and random sampling. Convergence to a degenerate diffusion process when N → ∞ is proved. We show how difference among alleles in means and variances of offspring numbers affects the limit diffusion, giving a rigorous multi-allelic version of a result of Gillespie.

Fractional Integrals and Extensions of Selfdecomposability

After characterizations of the class L of selfdecomposable distributions on \({\mathbb{R}}^{d}\) are recalled, the classes K p, α and L p, α with two continuous parameters 0 < p < ∞ and − ∞ < α < 2 satisfying \({K}_{1,0} = {L}_{1,0} = L\) are introduced as extensions of the class L. They are defined as the classes of distributions of improper stochastic integrals ∫0 ∞ − f(s)dX s (ρ), where f(s) is an appropriate non-random function and X s (ρ) is a Levy process on \({\mathbb{R}}^{d}\) with distribution ρ at time 1. The description of the classes is given by characterization of their Levy measures, using the notion of monotonicity of order p based on fractional integrals of measures, and in some cases by addition of the condition of zero mean or some weaker conditions that are newly introduced – having weak mean 0 or having weak mean 0 absolutely. The class L n, 0 for a positive integer n is the class of n times selfdecomposable distributions. Relations among the classes are studied. The limiting classes as p → ∞ are analyzed. The Thorin class T, the Goldie–Steutel–Bondesson class B, and the class L ∞ of completely selfdecomposable distributions, which is the closure (with respect to convolution and weak convergence) of the class \(\mathfrak{S}\) of all stable distributions, appear in this context. Some subclasses of the class L ∞ also appear. The theory of fractional integrals of measures is built. Many open questions are mentioned.

A class of multivariate infinitely divisible distributions related to arcsine density

Two transformations A1 and A2 of Levy measures on R d based on the arcsine density are studied and their relation to general Upsilon transformations is considered. The domains of definition of A1 and A2 are determined and it is shown that they have the same range. The class of infinitely divisible distributions on R d with Levy measures being in the common range is called the class A and any distribution in the class A is expressed as the law of a stochastic integral � 1 0 cos(2 −1 πt)dXt with respect to a Levy process {Xt }. This new class includes as a proper subclass the Jurek class of distributions. It is shown that generalized type G distributions are the image of distributions in the class A under a mapping defined by an appropriate stochastic integral. A2 is identified as an Upsilon transformation, while A1 is shown not to be.

Multidimensional process of Ornstein-Uhlenbeck type with nondiagonalizable matrix in linear drift terms

Let R d be the d -dimensional Euclidean space where each point is expressed by a column vector. Let | x | and ‹ x, y › denote the norm and the inner product in R d . Let Q = ( Q jk ) be a real d × d -matrix of which all eigenvalues have positive real parts. Let X be a process of Ornstein-Uhlenbeck type (OU type process) on R d associated with a Levy process {Z : t ≥ 0} and the matrix Q . Main purpose of this paper is to give a recurrence-transience criterion for the process X when Q is a Jordan cell matrix and to compare it with the case when Q is diagonalizable. Here by a Levy process we mean a stochastically continuous process with stationary independent increments, starting at 0.

SEMIGROUPS AND PROCESSES WITH PARAMETER IN A CONE

For a cone K in a Euclidean space recent results of Pedersen and Sato on K-parameter convolution semigroups and K-parameter Levy processes in law are surveyed. Relations to other multi-parameter processes are discussed. 1. Convolution semigroups with parameter in a cone A convolution semigroup with parameter in R+ = [0,∞), or an R+-parameter convolution semigroup, is a family of probability measures {μt : t ∈ R+} on R d such that (1.1) μt1 ∗ μt2 = μt1+t2 for t1, t2 ∈ R+