Variance-Mean Mixtures As Asymptotic Approximations

We present a general transfer theorem for 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 cases of random sums of independent not necessarily identically distributed random variables and statistics constructed from samples with random sizes. Using simple moment-type conditions we prove the theorem on convergence of the distributions of such sums to normal variance-mean mixtures.