Normal approximation for a randomly indexed branching process (Q6569672)

Let \((Z_n, n \geq 0)\) be a supercritical Galton-Watson process, and \((N(t), t \geq 0)\) an independent renewal process with inter-arrival time \(T\). The process \((Y_t = Z_{N(t)}, t \geq 0)\) is called a randomly indexed branching process (BRPI). Such a process has been proposed as an alternative to geometric Brownian motion for modelling stock prices.\N\NAssuming that the Galton-Watson process never dies out with \(m = \mathbb{E}(Z_1) \in (1,\infty)\) and satisfies the Kesten-Stygum integrability condition, that \(T\) is non-lattice and \(\mathbb E(T^3) < \infty\), the authors prove a one-term Edge-worth expansion for the asymptotic normality of \(Y\), i.e., that\N\[\N\sup_{x \in \mathbb R} \left| \mathbb P \left( \frac{\log Y_t - (\theta t - U) \log m}{\sigma \sqrt{t} \log m} \leq x \right) - \Phi(x) - \frac{Q(x) \phi(x)}{\sqrt{t}} \right| = o(t^{-1/2}) \text{ as \(t \to \infty\)},\N\]\Nwhere \(\theta,\sigma\) are explicit constants depending on the law of \(T\), \(U\) is an independent uniform random variable on \([0,1]\), \(Q\) is an explicit polynomial of degree \(2\) and \(\Phi\) (respectively \(\phi\)) is the cumulative distribution function (resp. probability distribution function) of the standard Gaussian distribution. In addition, the authors prove a Cramér-type moderate deviations for \(\log Y_t\), showing that its distribution is well approached, for \(x \in \theta t \pm o(t^{2/3})\) by a Gaussian random variable.\N\NThe results are illustrated with numerical simulations. The proofs rely on the use of similar Edge-worth expansion results known to hold for \((N(t), t \geq 0)\) found in [\textit{G. J. Babu} et al., Ann. Inst. Stat. Math. 55, No. 1, 83--94 (2003; Zbl 1050.62020)] as well as the a.s. convergence of \(m^{-n}Z_n\).