An asymptotic expansion for the expectation of an age-dependent branching process with a submultiplicative estimate of the remainder (Q2774461)

The aim of the paper is to investigate the asymptotic behavior of the expected number of particles \(\mu(t):= {\mathbf E}Z(t)\) as \(t \to \infty\). Here \(\{Z(t)\), \(t \geq 0\}\) is an age-dependent branching process with particles of one type. The process is governed by a generating function of offspring \(X\) of a particle, \(m ={\mathbf E}X < \infty\), and by a distribution \(F\) of the lifetime of a single particle. There is derived, among others, the asymptotic relation NEWLINE\[NEWLINE\mu(t) = c e^{\alpha t} + o(1/\varphi(t)) \quad\text{as }t \to \infty.NEWLINE\]NEWLINE Here \(\alpha \in R\) is the Malthusian parameter, i.e., \(m \int_0^\infty e^{-\alpha y}~F(dy)= 1\), \(c\) is a quantity depending on \(m\) and \(\alpha\), and \(\varphi\) is submultiplicative, i.e. a finite, positive, Borel measurable function such that \(\varphi(0)=1\), \(\varphi(x+y) \leq \varphi(x)\varphi(y)\), \(x,y \in R\). [The already known remainder term is \(o(e^{\alpha t})\).] There it is also proved a renewal theorem, emphasizing the influence of the roots of the characteristic equation on the asymptotic properties of a renewal measure (produced with the aid of iterated convolutions of \(F\), starting from the atomic measure of unit mass at the origin).