The simple branching process with infinite mean. I

The simple branching process {Z,) with mean number of offspring per individual infinite, is considered. Conditions under which there exists a sequence {p,} of positive constants such that p,, log (1 +Z,) converges in law to a proper limit distribution are given, as is a supplementary condition necessary and sufficient for p, constant cn as n-- oo, where 0 < c < 1 is a number characteristic of the process. Some properties of the limiting distribution function are discussed; while others (with additional results) are deferred to a sequel. BRANCHING PROCESS; GALTON-WATSON PROCESS; INFINITE MEAN; CONVEXITY;