arXiv subjects

Ken-iti Sato

Publications and source records attributed to Ken-iti Sato.

13 recordsLinked to original sources

arXiv2017

On quasi-infinitely divisible distributions

A quasi-infinitely divisible distribution on $\mathbb{R}$ is a probability distribution whose characteristic function allows a Lévy-Khintchine type representation with a "signed Lévy measure", rather than a Lévy measure. Quasi-infinitely divisible distributions appear naturally in the factorization of infinitely divisible distributions. Namely, a distribution $μ$ is quasi-infinitely divisible if and only if there are two infinitely divisible distributions $μ_1$ and $μ_2$ such that $μ_1 \ast μ= μ_2$. The present paper studies certain properties of quasi-infinitely divisible distributions in terms of their characteristic triplet, such as properties of supports, finiteness of moments, continuity properties and weak convergence, with various examples constructed. In particular, it is shown that the set of quasi-infinitely divisible distributions is dense in the set of all probability distributions with respect to weak convergence. Further, it is proved that a distribution concentrated on the integers is quasi-infinitely divisible if and only if its characteristic function does not have zeroes, with the use of the Wiener-Lévy theorem on absolutely convergent Fourier series. A number of fine properties of such distributions are proved based on this fact. A similar characterisation is not true for non-lattice probability distributions on the line.

math.PR↗

arXiv2012

A class of multivariate infinitely divisible distributions related to arcsine density

Two transformations $\mathcal{A}_1$ and $\mathcal{A}_2$ of Lévy measures on $\mathbb{R}^d$ based on the arcsine density are studied and their relation to general Upsilon transformations is considered. The domains of definition of $\mathcal{A}_1$ and $\mathcal{A}_2$ are determined and it is shown that they have the same range. The class of infinitely divisible distributions on $\mathbb{R}^d$ with Lévy 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 $\int_0^1\cos(2^{-1}\uppi t)\,\mathrm{d}X_t$ with respect to a Lévy process $\{X_t\}$. 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. $\mathcal{A}_2$ is identified as an Upsilon transformation, while $\mathcal{A}_1$ is shown not to be.

math.ST↗

arXiv2012

Inversions of infinitely divisible distributions and conjugates of stochastic integral mappings

The dual of an infinitely divisible distribution on $\mathbb{R}^d$ without Gaussian part defined in Sato, ALEA {\bf 3} (2007), 67--110, is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping $μ=Φ_{f}ρ$ of $ρ$ to $μ$ in the class of infinitely divisible distributions on $\mathbb{R}^d$, where $μ$ is the distribution of an improper stochastic integral of a nonrandom function $f$ with respect to a Lévy process on $\mathbb{R}^d$ with distribution $ρ$ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.

math.PR↗

arXiv2012

Weak drifts of infinitely divisible distributions and their applications

Weak drift of an infinitely divisible distribution $μ$ on $\mathbb{R}^d$ is defined by analogy with weak mean; properties and applications of weak drift are given. When $μ$ has no Gaussian part, the weak drift of $μ$ equals the minus of the weak mean of the inversion $μ'$ of $μ$. Applying the concepts of having weak drift 0 and of having weak drift 0 absolutely, the ranges, the absolute ranges, and the limit of the ranges of iterations are described for some stochastic integral mappings. For Lévy processes the concepts of weak mean and weak drift are helpful in giving necessary and sufficient conditions for the weak law of large numbers and for the weak version of Shtatland's theorem on the behavior near $t=0$; those conditions are obtained from each other through inversion.

math.PR↗

arXiv2011

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

Properties of the law $μ$ of the integral $\int_0^{\infty}c^{-N_{t-}}\,dY_t$ are studied, where $c>1$ and $\{(N_t,Y_t),t\geq0\}$ is a bivariate Lévy 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 Lévy 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 $r\leq pq$. 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 $μ$ is absolutely continuous whenever $q\geq C_1p\geq C_2r$. For any $c>1$ there is a positive constant $C_3$ such that $μ$ is continuous-singular whenever $q>0$ and $\max\{q,r\}\leq C_3p$. Here, if $\{N_t\}$ and $\{Y_t\}$ are independent, then $r=0$ and $q=b/(a+b)$.

math.PR↗

arXiv2010

Description of limits of ranges of iterations of stochastic integral mappings of infinitely divisible distributions

For infinitely divisible distributions $ρ$ on $\mathbb{R}^d$ the stochastic integral mapping $Φ_fρ$ is defined as the distribution of improper stochastic integral $\int_0^{\infty-} f(s) dX_s^{(ρ)}$, where $f(s)$ is a non-random function and $\{X_s^{(ρ)}\}$ is a Lévy process on $\mathbb{R}^d$ with distribution $ρ$ at time 1. For three families of functions $f$ with parameters, the limits of the nested sequences of the ranges of the iterations $Φ_f^n$ are shown to be some subclasses, with explicit description, of the class $L_{\infty}$ of completely selfdecomposable distributions. In the critical case of parameter 1, the notion of weak mean 0 plays an important role. Examples of $f$ with different limits of the ranges of $Φ_f^n$ are also given.

math.PR↗

arXiv2010

Type A Distributions: Infinitely Divisible Distributions Related to Arcsine Density

Two transformations $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ of Lévy measures on $\mathbb{R}^{d}$ based on the arcsine density are studied and their relation to general Upsilon transformations is considered. The domains of definition of $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are determined and it is shown that they have the same range. Infinitely divisible distributions on $\mathbb{R}^{d}$ with Lévy measures being in the common range are called type $A$ distributions and expressed as the law of a stochastic integral $\int_0^1\cos (2^{-1}πt)dX_t$ with respect to Lévy process $\{X_t\}$. \ 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 type $A$ distributions under a mapping defined by an appropriate stochastic integral. $\mathcal{A}_{2}$ is identified as an Upsilon transformation, while $\mathcal{A}_{1}$ is shown to be not.

math.PR↗

arXiv2009

Properties of stationary distributions of a sequence of generalized Ornstein-Uhlenbeck processes

The infinite (in both directions) sequence of the distributions $μ^{(k)}$ of the stochastic integrals $\int_0^{\infty-}c^{-N_{t-}^{(k)}} dL_t^{(k)}$ for integers $k$ is investigated. Here $c>1$ and $(N_t^{(k)},L_t^{(k)})$, $t\geq0$, is a bivariate compound Poisson process with Lévy measure concentrated on three points $(1,0)$, $(0,1)$, $(1,c^{-k})$. The amounts of the normalized Lévy measure at these points are denoted by $p$, $q$, $r$. For $k=0$ the process $(N_t^{(0)},L_t^{(0)})$ is marginally Poisson and $μ^{(0)}$ has been studied by Lindner and Sato (Ann. Probab. 37 (2009), 250-274). The distributions $μ^{(k)}$ are the stationary distributions of a sequence of generalized Ornstein-Uhlenbeck processes structurally related in some way. Continuity properties of $μ^{(k)}$ are shown to be the same as those of $μ^{(0)}$. The problem to find necessary and sufficient conditions in terms of $c$, $p$, $q$, and $r$ for $μ^{(k)}$ to be infinitely divisible is somewhat involved, but completely solved for every integer $k$. The conditions depend on arithmetical properties of $c$. The symmetrizations of $μ^{(k)}$ are also studied. The distributions $μ^{(k)}$ and their symmetrizations are $c^{-1}$-decomposable, and it is shown that, for each $k\neq 0$, $μ^{(k)}$ and its symmetrization may be infinitely divisible without the corresponding factor in the $c^{-1}$-decomposability relation being infinitely divisible. This phenomenon was first observed by Niedbalska-Rajba (Colloq. Math. 44 (1981), 347-358) in an artificial example. The notion of quasi-infinite divisibility is introduced and utilized, and it is shown that a quasi-infinitely divisible distribution on $[0,\infty)$ can have its quasi-Lévy measure concentrated on $(-\infty,0)$.

math.PR↗

arXiv2007

The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions

It is shown that the limits of the nested subclasses of five classes of infinitely divisible distributions on $R^d$, which are the Jurek class, the Goldie-Steutel-Bondesson class, the class of selfdecomposable distributions, the Thorin class and the class of generalized type $G$ distributions, are identical with the closure of the class of stable distributions. More general results are also given.

math.PR↗

arXiv2007

Selfdecomposability and semi-selfdecomposability in subordination of cone-parameter convolution semigroups

Extension of two known facts concerning subordination is made. The first fact is that, in subordination of 1-dimensional Brownian motion with drift, selfdecomposability is inherited from subordinator to subordinated. This is extended to subordination of cone-parameter convolution semigroups. The second fact is that, in subordination of strictly stable cone-parameter convolution semigroups on $\mathbb{R}^d$, selfdecomposability is inherited from subordinator to subordinated. This is extended to semi-selfdecomposability.

math.PR↗

arXiv2007

Transformations of infinitely divisible distributions via improper stochastic integrals

Let $X^{(μ)}(ds)$ be an $\mathbb{R}^d$-valued homogeneous independently scattered random measure over $\mathbb{R}$ having $μ$ as the distribution of $X^{(μ)}((t,t+1])$. Let $f(s)$ be a nonrandom measurable function on an open interval $(a,b)$ where $-\infty\leqslant a<b\leqslant\infty$. The improper stochastic integral $\int_{a+}^{b-} f(s)X^{(μ)}(ds)$ is studied. Its distribution $Φ_f(μ)$ defines a mapping from $μ$ to an infinitely divisible distribution on $\mathbb{R}^d$. Three modifications (compensated, essential, and symmetrized) and absolute definability are considered. After their domains are characterized, necessary and sufficient conditions for the domains to be very large (or very small) in various senses are given. The concept of the dual in the class of purely non-Gaussian infinitely divisible distributions on $\mathbb{R}^d$ is introduced and employed in studying some examples. The $τ$-measure $τ$ of function $f$ is introduced and whether $τ$ determines $Φ_f$ is discussed. Related transformations of Lévy measures are also studied.

math.PR↗

arXiv2006

Monotonicity and non-monotonicity of domains of stochastic integral operators

A Lévy process on $R^d$ with distribution $μ$ at time 1 is denoted by $X^{(μ)}=\{X_t^{(μ)}\}$. If the improper stochastic integral $\int_0^{\infty-} f(s)dX_s^{(μ)}$ of $f$ with respect to $X^{(μ)}$ is definable, its distribution is denoted by $Φ_f(μ)$. The class of all infinitely divisible distributions $μ$ on $R^d$ such that $Φ_f(μ)$ is definable is denoted by $D(Φ_f)$. The class $D(Φ_f)$, its two extensions $D_c(Φ_f)$ and $D_e(Φ_f)$ (compensated and essential), and its restriction $D^0(Φ_f)$ (absolutely definable) are studied. It is shown that $D_e(Φ_f)$ is monotonic with respect to $f$, which means that $|f_2|\leq |f_1|$ implies $D_e(Φ_{f_1})\subset D_e(Φ_{f_2})$. Further, $D^0(Φ_f)$ is monotonic with respect to $f$ but neither $D(Φ_f)$ nor $D_c(Φ_f)$ is monotonic with respect to $f$. Furthermore, there exist $μ$, $f_1$, and $f_2$ such that $0\leq f_2\leq f_1$, $μ\in D(Φ_{f_1})$, and $μ\not\in D(Φ_{f_2})$. An explicit example for this is related to some properties of a class of martingale Lévy processes.

math.PR↗

arXiv2006

Some properties of exponential integrals of Lévy processes and examples

The improper stochastic integral $Z=\int_0^{\infty-}\exp(-X_{s-})dY_s$ is studied, where $\{(X_t, Y_t), t \geqslant 0 \}$ is a Lévy process on $\mathbb R ^{1+d}$ with $\{X_t \}$ and $\{Y_t \}$ being $\mathbb R$-valued and $\mathbb R ^d$-valued, respectively. The condition for existence and finiteness of $Z$ is given and then the law $\mathcal L(Z)$ of $Z$ is considered. Some sufficient conditions for $\mathcal L(Z)$ to be selfdecomposable and some sufficient conditions for $\mathcal L(Z)$ to be non-selfdecomposable but semi-selfdecomposable are given. Attention is paid to the case where $d=1$, $\{X_t\}$ is a Poisson process, and $\{X_t\}$ and $\{Y_t\}$ are independent. An example of $Z$ of type $G$ with selfdecomposable mixing distribution is given.

math.PR↗