On normal variance–mean mixtures as limit laws for statistics with random sample sizes

• We prove a general transfer theorem for multivariate random sequences with independent random indexes in the double array limit setting.
• 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 multivariate normal variance–mean mixtures and, in particular, to multivariate generalized hyperbolic laws.

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 multivariate normal variance–mean mixtures and, in particular, to multivariate generalized hyperbolic laws.