Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| author | Iksanov, Alexander Kolesko, Konrad Meiners, Matthias |
| author_facet | Iksanov, Alexander Kolesko, Konrad Meiners, Matthias |
| contents | Consider a supercritical Crump--Mode--Jagers process $(\mathcal Z_t^φ)_{t \geq 0}$ counted with a random characteristic $φ$. Nerman's celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365--395, 1981] states that, under some mild assumptions, $e^{-αt} \mathcal Z_t^φ$ converges almost surely as $t \to \infty$ to $aW$. Here, $α>0$ is the Malthusian parameter, $a$ is a constant and $W$ is the limit of Nerman's martingale, which is positive on the survival event. In this general situation, under additional (second moment) assumptions, we prove a central limit theorem for $(\mathcal Z_t^φ)_{t \geq 0}$. More precisely, we show that there exist a constant $k \in \mathbb N_0$ and a function $H(t)$, a finite random linear combination of functions of the form $t^j e^{λt}$ with $α/2 \leq \mathrm{Re}(λ)<α$, such that $(\mathcal Z_t^φ- a e^{αt}W -H(t))/\sqrt{t^k e^{αt}}$ converges in distribution to a normal random variable with random variance. This result unifies and extends various central limit theorem-type results for specific branching processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_00867 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes Iksanov, Alexander Kolesko, Konrad Meiners, Matthias Probability 60J80 (Primary) 60F05, 60G44 (Secondary) Consider a supercritical Crump--Mode--Jagers process $(\mathcal Z_t^φ)_{t \geq 0}$ counted with a random characteristic $φ$. Nerman's celebrated law of large numbers [Z. Wahrsch. Verw. Gebiete 57, 365--395, 1981] states that, under some mild assumptions, $e^{-αt} \mathcal Z_t^φ$ converges almost surely as $t \to \infty$ to $aW$. Here, $α>0$ is the Malthusian parameter, $a$ is a constant and $W$ is the limit of Nerman's martingale, which is positive on the survival event. In this general situation, under additional (second moment) assumptions, we prove a central limit theorem for $(\mathcal Z_t^φ)_{t \geq 0}$. More precisely, we show that there exist a constant $k \in \mathbb N_0$ and a function $H(t)$, a finite random linear combination of functions of the form $t^j e^{λt}$ with $α/2 \leq \mathrm{Re}(λ)<α$, such that $(\mathcal Z_t^φ- a e^{αt}W -H(t))/\sqrt{t^k e^{αt}}$ converges in distribution to a normal random variable with random variance. This result unifies and extends various central limit theorem-type results for specific branching processes. |
| title | Asymptotic fluctuations in supercritical Crump-Mode-Jagers processes |
| topic | Probability 60J80 (Primary) 60F05, 60G44 (Secondary) |
| url | https://arxiv.org/abs/2109.00867 |