An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process

Standard

An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process. / Tsilevich, Natalia ; Vershik, Anatoly; Yor, Marc.

In: Journal of Functional Analysis, Vol. 185, No. 1, 10.09.2001, p. 274-296.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{4c4bf8e16467453cb4cff7550ede7475,

title = "An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process",

abstract = "We define a one-parameter family ℒχ of sigma-finite (finite on compact sets) measures in the space of distributions. These measures are equivalent to the laws of the classical gamma processes and invariant under an infinite-dimensional abelian group of certain positive multiplicators. This family of measures was first discovered by Gelfand-Graev-Vershik in the context of the representation theory of current groups; here we describe it in direct terms using some remarkable properties of the gamma processes. We show that the class of multiplicative measures coincides with the class of zero-stable measures which is introduced in the paper. We give also a new construction of the canonical representation of the current group SL(2,ℝ)X.",

keywords = "Gamma process, Infinite-dimensional Lebesgue measure, Sigma-finite invariant zero-stable measures",

author = "Natalia Tsilevich and Anatoly Vershik and Marc Yor",

year = "2001",

month = sep,

day = "10",

doi = "10.1006/jfan.2001.3767",

language = "English",

volume = "185",

pages = "274--296",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Elsevier",

number = "1",

}

RIS

TY - JOUR

T1 - An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process

AU - Tsilevich, Natalia

AU - Vershik, Anatoly

AU - Yor, Marc

PY - 2001/9/10

Y1 - 2001/9/10

N2 - We define a one-parameter family ℒχ of sigma-finite (finite on compact sets) measures in the space of distributions. These measures are equivalent to the laws of the classical gamma processes and invariant under an infinite-dimensional abelian group of certain positive multiplicators. This family of measures was first discovered by Gelfand-Graev-Vershik in the context of the representation theory of current groups; here we describe it in direct terms using some remarkable properties of the gamma processes. We show that the class of multiplicative measures coincides with the class of zero-stable measures which is introduced in the paper. We give also a new construction of the canonical representation of the current group SL(2,ℝ)X.

AB - We define a one-parameter family ℒχ of sigma-finite (finite on compact sets) measures in the space of distributions. These measures are equivalent to the laws of the classical gamma processes and invariant under an infinite-dimensional abelian group of certain positive multiplicators. This family of measures was first discovered by Gelfand-Graev-Vershik in the context of the representation theory of current groups; here we describe it in direct terms using some remarkable properties of the gamma processes. We show that the class of multiplicative measures coincides with the class of zero-stable measures which is introduced in the paper. We give also a new construction of the canonical representation of the current group SL(2,ℝ)X.

KW - Gamma process

KW - Infinite-dimensional Lebesgue measure

KW - Sigma-finite invariant zero-stable measures

UR - http://www.scopus.com/inward/record.url?scp=0035840456&partnerID=8YFLogxK

U2 - 10.1006/jfan.2001.3767

DO - 10.1006/jfan.2001.3767

M3 - Article

AN - SCOPUS:0035840456

VL - 185

SP - 274

EP - 296

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -

ID: 49790219