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.