A universal right tail upper bound for supercritical Galton-Watson processes with bounded offspring
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916087814160384 |
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| author | Fernley, John Jacob, Emmanuel |
| author_facet | Fernley, John Jacob, Emmanuel |
| contents | We consider a supercritical Galton-Watson process $Z_n$ whose offspring distribution has mean $m>1$ and is bounded by some $d\in \{2,3,\ldots\}$. As well-known, the associated martingale $W_n=Z_n/m^n$ converges a.s. to some nonnegative random variable $W_\infty$. We provide a universal upper bound for the right tail of $W_\infty$ and $W_n$, which is uniform in $n$ and in all offspring distributions with given $m$ and $d$, namely:
\[
P(W_n\ge x)\le c_1 \exp\left\{-c_2 \frac {m-1}m \frac x d\right\}, \quad \forall n\in \mathbb N \cup \{+\infty\}, \forall x\ge 0,
\]
for some explicit constants $c_1,c_2>0$. For a given offspring distribution, our upper bound decays exponentially as $x\to \infty$, which is actually suboptimal, but our bound is $\textit{universal}$: it provides a single $\textit{effective}$ expression, which is $\textit{nonasymptotic}$ - it does not require $x$ large - and valid simultaneously for all supercritical bounded offspring distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07241 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A universal right tail upper bound for supercritical Galton-Watson processes with bounded offspring Fernley, John Jacob, Emmanuel Probability 60J80 (Primary) 60F10 (Secondary) We consider a supercritical Galton-Watson process $Z_n$ whose offspring distribution has mean $m>1$ and is bounded by some $d\in \{2,3,\ldots\}$. As well-known, the associated martingale $W_n=Z_n/m^n$ converges a.s. to some nonnegative random variable $W_\infty$. We provide a universal upper bound for the right tail of $W_\infty$ and $W_n$, which is uniform in $n$ and in all offspring distributions with given $m$ and $d$, namely: \[ P(W_n\ge x)\le c_1 \exp\left\{-c_2 \frac {m-1}m \frac x d\right\}, \quad \forall n\in \mathbb N \cup \{+\infty\}, \forall x\ge 0, \] for some explicit constants $c_1,c_2>0$. For a given offspring distribution, our upper bound decays exponentially as $x\to \infty$, which is actually suboptimal, but our bound is $\textit{universal}$: it provides a single $\textit{effective}$ expression, which is $\textit{nonasymptotic}$ - it does not require $x$ large - and valid simultaneously for all supercritical bounded offspring distributions. |
| title | A universal right tail upper bound for supercritical Galton-Watson processes with bounded offspring |
| topic | Probability 60J80 (Primary) 60F10 (Secondary) |
| url | https://arxiv.org/abs/2307.07241 |