arXiv subjects

Toshiro Watanabe

Publications and source records attributed to Toshiro Watanabe.

11 recordsLinked to original sources

arXiv2023

On convolution closure properties of subexponentiality approaching from densities

Non-closedness of subexponentiality by the convolution operation is well-known. We go a step further and show that subexponentiality and non-subexponentiality are generally changeable by the convolution. We also give several conditions, by which (non-) subexponentiality is kept. Most results are given with densities, which are easily converted to those for distributions. As a by-product, we give counterexamples to several past results, which were used to derive the non-closedness of the convolution, and modify the original proof.

math.PR↗

arXiv2023

Local subexponentiality and infinitely divisible distributions

We completely characterize $Δ$- and local subexponentialities of positive-half compound Poisson distributions and extend the characterization on two-sided distributions. Moreover, $Δ$-subexponentiality of infinitely divisible distributions is characterized with new conditions, and local subexponentiality is newly characterized in the two-sided case. In the process closedness properties of these subexponentialities are derived, particularly for distributions on $\R$. Most results are obtained by exploiting monotonic-type assumptions. We apply our results to distributions of supremum of a random work and a randomly stopped iid sum.

math.PR↗

arXiv2020

Second order subexponentiality and infinite divisibility

We characterize the second order subexponentiality of an infinitely divisible distribution on the real line under an exponential moment assumption. We investigate the asymptotic behaviour of the difference between the tails of an infinitely divisible distribution and its Lévy measure. Moreover, we study the second order asymptotic behaviour of the tail of the $t$-th convolution power of an infinitely divisible distribution. The density version for a self-decomposable distribution on the real line without an exponential moment assumption is also given. Finally, the regularly varying case for a self-decomposable distribution on the half line is discussed.

math.PR↗

arXiv2020

Subexponential densities of compound Poisson sums and the supremum of a random walk

We characterize the subexponential densities on $(0,\infty)$ for compound Poisson distributions on $[0,\infty)$ with absolutely continuous Lévy measures. As a corollary, we show that the class of all subexponential probability density functions on $\mathbb R_+$ is closed under generalized convolution roots of compound Poisson sums. Moreover, we give an application to the subexponential density on $(0,\infty)$ for the distribution of the supremum of a random walk.

math.PR↗

arXiv2019

Two hypotheses on the exponential class in the class of $O$-subexponential infinitely divisible distributions

Two hypotheses on the class $\mathcal{L}(γ)$ in the class $\mathcal{OS}\cap \mathcal{ID}$ are discussed. Two weak hypotheses on the class $\mathcal{L}(γ)$ in the class $\mathcal{OS}\cap \mathcal{ID}$ are proved. A necessary and sufficient condition in order that, for every $t >0$, the $t$-th convolution power of a distribution in the class $\mathcal{OS}\cap \mathcal{ID}$ belongs to the class $\mathcal{L}(γ)$ is given.

math.PR↗

arXiv2019

Subexponential densities of infinitely divisible distributions on the half line

We show that, under the long-tailedness of the densities of normalized Lévy measures, the densities of infinitely divisible distributions on the half line are subexponential if and only if the densities of their normalized Lévy measures are subexponential. Moreover, we prove that, under a certain continuity assumption, the densities of infinitely divisible distributions on the half line are subexponential if and only if their normalized Lévy measures are locally subexponential.

math.PR↗

arXiv2016

Two non-closure properties on the class of subexponential densities

Relations between subexponential densities and locally subexponential distributions are discussed. It is shown that the class of subexponential densities is neither closed under convolution roots nor closed under asymptotic equivalence. A remark is given on the closure under convolution roots for the class of convolution equivalent distributions.

math.PR↗

arXiv2015

The conjectures of Embrechts and Goldie

It is shown that the class of convolution equivalent distributions and the class of locally subexponential distributions are not closed under convolution roots. Moreover, two sufficient conditions for the closure under convolution roots of the class of convolution equivalent distributions are given.

math.PR↗

arXiv2014

Escape rates for multi-dimensional shift selfsimilar additive sequences

First the relation between shift selfsimilar additive sequences and stationary sequences of Ornstein-Uhlenbeck type (OU type) on $\mathbb{R}^d$ is shown and then the rates of escape for shift selfsimilar additive sequences are discussed. As a corollary, fundamental problems on recurrence of stationary sequences of OU type are solved. Some applications to laws of the iterated logarithm for strictly stable Lévy processes on $\mathbb{R}^d$ and independent Brownian motions are given.

math.PR↗

arXiv2010

Limsup behaviors of multi-dimensional selfsimilar processes with independent increments

Laws of the iterated logarithm of "limsup" type are studied for multi-dimensional selfsimilar processes $\{X(t)\}$ with independent increments having exponent $H$. It is proved that, for any positive increasing function $g(t)$ with $\displaystyle \lim_{t\to\infty}g(t) = \infty$, there is $C\in [0,\infty]$ such that $\limsup|X(t)|/(t^Hg(|\log t|))= C$ a.s. as $t \to\infty $, in addition, as $t \to 0$. A necessary and sufficient condition for the existence of $g(t)$ with C=1 is obtained. In the case where $g(t)$ with C=1 does not exist, a criterion to classify functions $g(t)$ according to C=0 or $C=\infty$ is given. Moreover, various "limsup" type laws with identification of the positive constants $C$ are explicitly presented in several propositions and examples. The problems that exchange the roles of $\{X(t)\}$ and $g(t)$ are also discussed.

math.PR↗

arXiv2007

Exact Hausdorff measure on the boundary of a Galton--Watson tree

A necessary and sufficient condition for the almost sure existence of an absolutely continuous (with respect to the branching measure) exact Hausdorff measure on the boundary of a Galton--Watson tree is obtained. In the case where the absolutely continuous exact Hausdorff measure does not exist almost surely, a criterion which classifies gauge functions $ϕ$ according to whether $ϕ$-Hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given. Important examples are discussed in four additional theorems. In particular, Hawkes's conjecture in 1981 is solved. Problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved.

math.PR↗