2007/03/10 by A. M. Vershik, Anatoly Vershik, Vershik, Anatoly · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #22E45 #46G12 #46G20 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and financial applications #math-ph #math.MP #math.PR #msc:22E45 #msc:46G12 #msc:46G20
paper · pdf · doi:10.48550/arxiv.math-ph/0703033
35 pp. Ref 39
openalex publication_date 2007/03/10 · arxiv created 2008/02/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the sigma-finite measures in the space of vector-valued distributions on the manifold X with Laplace transform Ψ(f)=exp\-θ∫Xln||f(x)||dx\, θ>0. We also consider the weak limit of Haar measures on the Cartan subgroup of the group SL(n,\Bbb R) when n tends to infinity. The measure in the limit is called \it infinite dimensional Lebesgue measure. It is invariant under the linear action of some infinite-dimensional Abelian group which is an analog of Cartan subgroup. The measure also is closely related to the Poisson--Dirichlet measures well known in combinatorics and probability theory. The only known example of the analogous asymptotical behavior of the uniform measure on the homogeneous manifold is \it classical Maxwell-Poincaré lemma which asserts that the weak limit of uniform measures on the Euclidean sphere of appropriate radius as dimension tends to infinity is the standard infinite-dimensional Gaussian measure and white noise, but in our situation all the measures are no more finite but sigma-finite.