  We prove a general transfer theorem for multivariate random sequences with
independent random indexes in the double array limit setting. We also prove its
partial inverse providing necessary and sufficient conditions for the
convergence of randomly indexed random sequences. Special attention is paid to
the case where the elements of the basic double array are formed as statistics
constructed from samples with random sizes. Under rather natural conditions we
prove the theorem on convergence of the distributions of such statistics to
normal variance-mean mixtures.
