It is shown that there is a close relationship between the convergence of a sequence of normalized Galton-Watson processes and the convergence of the rowsums of a certain triangular array of independent identically distributed random variables. Using this result some limit theorems by Jirina and Lamperti are strengthened.
"On the Convergence of Sequences of Branching Processes." Ann. Probab. 2 (6) 1027 - 1045, December, 1974. https://doi.org/10.1214/aop/1176996496