  We prove a version of a general transfer theorem for random sequences with
independent random indexes in the double array limit setting under relaxed
conditions. We also prove its partial inverse providing the 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 cumulative sums of independent not necessarily identically
distributed random variables. Using simple moment-type conditions we prove the
theorem on convergence of the distributions of such sums to normal
variance-mean mixtures.
