Chris Heyde on Branching Processes and Population Genetics

Abstract

this commentary. A strong impetus to branching process theory had occurred with the appearance of the book of Harris (1963).The dominant theme of Chris's work prior to his arrival had been refinement of classical limit theory for sums of independent random variables. A principal theme of his work since 1970 has been the theory and use of martingale methods in situations of statistical dependence. This seems to have its origin in his paper [M31], in the context of a simple (Bienaymé-Galton-Watson) branching process {Z n }, n ³ 0; Z 0 = 1. Suppose the non-degenerate offspring distribution has probability generating function< ∞ the process is called supercritical, and in this case the probability of extinction, q, is the unique real number in [0, 1) satisfying F(q) = q. The random sequence {Z n ∕ m n } is clearly a non-negative martingale, which in the supercritical case under the-then standard condition 2 1( ) E Z < ∞ was known to converge almost surely (a.s) to a random variable W for which P(W = 0) = q and which has a continuous density on the set of positive real numbers. After a number of refinements of this convergence result, it was shown in Seneta (1968) that in the non-degenerate supercitical case without further conditions, the choice c n = 1 ∕ h n (s 0 ), where s 0 is any fixed number in (0, − logq), and h n (s 0 ) is the n-th functional iterate of h(s), which is the inverse function of k(s) = − log F (e − s ), results in convergence in distribution of {Z n ∕ c n } to a proper non-degenerate random variable, W. While it was noticed that {W n = h n (s 0 )Z n } was a submartingale, a.s. convergence could not be deduced from the usual theorem since E(W n ) is bounded if and only if E(Z 1 logZ 1 ) < ∞. Chris's insight was that {exp ( − h n (s 0 )Z n )} is a martingale, with (obviously) bounded mean, thus giving the sharpening to a.s. convergence. Reciprocally, this strengthening immediately led to an analogous result for the supercritical branching process with immigration {X n } (Seneta, 1970)