Limit theorems for random sums of dependent d-dimensional random vectors

Abstract

Let there be given a doubly infinite sequence ~ = { { X , k ; k e N } n e N } of ddimensional random vectors (rv's) defined on a common probability space 4 " (f2,~,,P) and a doubly infinite sequence 5 = { { ,k, ke{0}wN};neN} of o-subfields of ~, which is row-wise increasing ~ a ~1) and adapted to \~ n,k ~ n,k+ 9 , i.e. every X,k is ~,k-measurable. A pair (~, 5) will be called a system. Moreover, let us choose a fixed sequence of random variables vm: (f2, o ~, P)-~ N u { + o o } , meN, such that every v m is a stopping time with respect to the m-th row of ~, i.e. [vmNk]e~m, 1, k e N . Finally, let us define a sequence of random sums: v,