An Infinite-Dimensional Analogue of the Lebesgue Measure and Distinguished Properties of the Gamma Process

✍ Scribed by Natalia Tsilevich; Anatoly Vershik; Marc Yor

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 366 KB
• Volume: 185
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3767

✦ Synopsis

We define a one-parameter family L h 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 .