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We prove a version of a general transfer theorem for random sequences withindependent random indexes in the double array limit setting under relaxedconditions. We also prove its partial inverse providing the necessary andsufficient conditions for the convergence of randomly indexed random sequences.Special attention is paid to the case where the elements of the basic doublearray are formed as cumulative sums of independent not necessarily identicallydistributed random variables. Using simple moment-type conditions we prove thetheorem on convergence of the distributions of such sums to normalvariance-mean mixtures.

On convergence of the distributions of random sequences with independent random indexes to variance-mean mixtures