Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups

Wilfried Hazod, Eberhard Siebert, "Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups: Structural Properties and Limit Theorems"
English | 2001 | pages: 629 | ISBN: 904815832X, 1402000405 | DJVU | 4,8 mb
Generalising classical concepts of probability theory, the investigation of operator (semi)-stable laws as possible limit distributions of operator-normalized sums of i.i.d. random variable on finite-dimensional vector space started in 1969. Currently, this theory is still in progress and promises interesting applications. Parallel to this, similar stability concepts for probabilities on groups were developed during recent decades. It turns out that the existence of suitable limit distributions has a strong impact on the structure of both the normalizing automorphisms and the underlying group. Indeed, investigations in limit laws led to contractable groups and - at least within the class of connected groups - to homogeneous groups, in particular to groups that are topologically isomorphic to a vector space. Moreover, it has been shown that (semi)-stable measures on groups have a vector space counterpart and vice versa.