The non-linear renewal equation

Asymptotic probabilities in a critical age-dependent branching process

“…• 0 A Newton binomial expansion applied to the rhs integrand of (2.21) using (2.10), Theorem 4, p. 406 of [ 3] , (2.16), and that ~ ~ 1 yields that the rha of (2.21) equals…”

The asymptotic behavior of a Volterra-renewal equation

“…Proceeding along the lines developed for the branching process in [3], one may, under stronger differentiability and moment conditions on h and G, refine the result of the theorem. For example, under very slightly more than the existence of h'"(0 +), and a fourth moment of G, one can show that *"'(') -Y_1r + y, logí + y2 + 0(t~y}), where y is as before, and yx, y2, y3 are constants, y, depending on h"(0), h'"(0), and p. We will not go into such refinements here.…”

“…In the applications, the proofs of several classical limit theorems rest crucially on determination of the rate of convergence of x(t) to L. To that end we initiated study of these rates with J. Chover [3] for the simplest branching model leading to (1), using techniques which depended on special aspects of the branching process.…”

“…(ii) h is a nonnegative, Lipschitz continuous function on a closed interval I containing the range of x, such that(2) (3) A(0) = 0, A'(0 +) = 1, A"(0 +) < 0;…”

On a multi-type critical age-dependent branching process

“…We will be interested in the quantities P,(t) of (2.3). It seems of independent interest to give a proof of (3.3) based on the corresponding result in the one-type case given by Chover and Ney [1]. It seems of independent interest to give a proof of (3.3) based on the corresponding result in the one-type case given by Chover and Ney [1].…”