This paper considers the general branching process counted by random characteristics, details of which can be found in the first author’s book, Branching processes with biological applications. (1975;

Zbl 0356.60039). Each individual x, with birth time $\sigma\sb x$, has associated with it a random function, called a characteristic, $\chi\sb x(u)$. Results on the convergence of $e\sp{-\alpha t}\sum\sb{x}\chi\sb x(t-\sigma\sb x)$ with $\alpha$ the Malthusian parameter have, at least for particular choices of $\chi$, a long history culminating in the paper by the second author, Z. Wahrscheinlichkeitstheor. Verw. Geb. 57, 365-395 (1981;

Zbl 0451.60078).
Here the framework is extended somewhat so that each individual has associated with it a whole collection of characteristics, one for each $t\ge 0$, $\{\chi\sb{t,x}:t\ge 0\}$ and $\sum\sb{x}\chi\sb{t,x}(t- \sigma\sb x)$ is considered as $t\to \infty$. This can be thought of as the row sums of a triangular array of characteristics. Suitable conditions on the collection of characteristics allow the convergence in distribution of $\sum\sb{x}\chi\sb{t,x}(t-\sigma\sb x)$ as $t\to \infty$ to be established. Also, as the authors point out, for the result obtained the important feature of the underlying process leading to the birth times is not that it is a general branching process but that it grows exponentially.
A wide variety of cases are covered by the theorem obtained and various special cases are discussed. A number of these relate to various kinds of rare events when the limits obtained are mixed Poisson. Others relate to the random error of population projections when the limits are mixed normal distributions (i.e. normals with a random variance). The main result also holds for the lattice case of the general branching process. For the supercritical Galton-Watson process one of the authors’ population projection examples corresponds to a theorem of {\it C. C. Heyde} on the rate of convergence in this process; see J. Appl. Probab. 7, 451-454 (1970;

Zbl 0198.225).