Limit theorems on the direct product of a non-compact Lie group and a compact group

Authors
Citation
P. Major et G. Pap, Limit theorems on the direct product of a non-compact Lie group and a compact group, ST SCI M H, 38, 2001, pp. 279-297
Citations number
8
Categorie Soggetti
Mathematics
Journal title
STUDIA SCIENTIARUM MATHEMATICARUM HUNGARICA
ISSN journal
00816906 → ACNP
Volume
38
Year of publication
2001
Pages
279 - 297
Database
ISI
SICI code
0081-6906(2001)38:<279:LTOTDP>2.0.ZU;2-N
Abstract
Let us consider a triangular array of random vectors (X-j((n)) , Y-j((n))), n = 1,2,..., 1 less than or equal to j less than or equal to k(n), such th at the first coordinates X-j((n)) take their values in a non-compact Lie gr oup and the second coordinates Y-j((n)) in a compact group. Let the random vectors (X-j((n)) , Y-j((n))) be independent for fixed n, but we do not ass ume any (independence type) condition about the relation between the compon ents of these vectors. We show under Icn kn fairly general conditions that if both random products S-n = Pi (kn)(j=1) X-j((n)) and Tn = Pi (kn)(j=1) Y -j((n)) have a limit distribution, then also the random vectors (S-n,T-n) c onverge in distribution as n --> infinity. Moreover, the non-compact and co mpact coordinates of a random vector with this limit distribution are indep endent.