Searcharxiv⌕ Search

arXiv subjects

Takahiro Hasebe

Publications and source records attributed to Takahiro Hasebe.

62 records · Page 4Linked to original sources
arXiv2011

Conditionally monotone independence I: Independence, additive convolutions and related convolutions

We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in non-commutative probability theory and unifies the monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the additive convolution and deformations of the monotone convolution. We define cumulants for a general convolution to analyze the deformed convolutions.

math.OA↗
arXiv2011

Conditionally monotone independence II: Multiplicative convolutions and infinite divisibility

We study the multiplicative convolution for c-monotone independence. This convolution unifies the monotone, Boolean and orthogonal multiplicative convolutions. We characterize convolution semigroups for the c-monotone multiplicative convolution on the unit circle. We also prove that an infinitely divisible distribution can always be embedded in a convolution semigroup. We furthermore discuss the (non)-uniqueness of such embeddings including the monotone case. Finally connections to the multiplicative Boolean convolution are discussed.

math.OA↗
arXiv2011

Analytic continuations of Fourier and Stieltjes transforms and generalized moments of probability measures

We consider analytic continuations of Fourier transforms and Stieltjes transforms. This enables us to define what we call complex moments for some class of probability measures which do not have moments in the usual sense. There are two ways to generalize moments accordingly to Fourier and Stieltjes transforms; however these two turn out to coincide. As applications, we give short proofs of the convergence of probability measures to Cauchy distributions with respect to tensor, free, Boolean and monotone convolutions.

math.PR↗
arXiv2011

White noise analysis on manifolds and the energy representation of a gauge group

The energy representation of a gauge group on a Riemannian manifold has been discussed by several authors. Y. Shimada has shown the irreducibility for compact Riemannian manifold, using white noise analysis. In this paper we extend its technique to noncompact Riemannian manifolds which have differential operators satisfying some conditions.

math-ph↗
arXiv2010

On monotone convolution and monotone infinite divisivility

This article focuses on properties of monotone convolutions. A criterion for infinite divisibility and time evolution of convolution semigroups are mainly studied. In particular, we clarify that many analogues of the classical results of Lévy processes hold such as characterizations of subordinators and strictly stable distributions.

math.OA↗
arXiv2010

The Monotone Cumulants

In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean, and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in the commutative, free and Boolean cases. The way we define (generalized) cumulants needs neither partition lattices nor generating functions and then will give a new viewpoint to cumulants. We define ``monotone cumulants'' in the sense of generalized cumulants and we obtain quite simple proofs of central limit theorem and Poisson's law of small numbers in monotone probability theory. Moreover, we clarify a combinatorial structure of moment-cumulant formula with the use of ``monotone partitions''.

math.PR↗