Limit theorems for statistics with random sample sizes
M. E. Grigoryeva, Victor Yu. Korolev, Alexander I. Zeifman · AIP conference proceedings · 2015
We prove an improved version of 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 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 sums to normal variance-mean mixtures.