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Letting Z(t) be the number of objects alive at time t in a general supercritical age-dependent branching process generated by a single ancestor born at time 0, one achieves (Theorem 1) mean-square convergence of Z(t)/E[Z(t)] provided 
and 
, where N(t) is the number of offspring of the initial ancestor born by time t and α is the (positive) Malthusian parameter defined by 
. If the stronger conditions that (Theorem 2) 
and 
hold also, then the convergence is almost-sure. It is of interest that the embedded Galton-Watson process of successive generations need not have a finite mean for the conditions of the above theorems to hold. Similar results are obtained for the age-distribution as well.
