Branching Process in a Varying Environment: How to Grow Like the Product of Means
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866913011947536384 |
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| author | Kirpicheva, Y. Shklyaev, A. |
| author_facet | Kirpicheva, Y. Shklyaev, A. |
| contents | Consider a branching process $\{Z_n\}$ in a varying environment. Let $\{W_n\}$ be the natural martingale $Z_n/{\bf E}Z_n$. It converges to some random variable $W$ as $n\to\infty$. An important problem is to show that ${\bf P}(W>0)$ equals the survival probability, so that $Z_n$ is either $0$ or of the order ${\bf E}Z_n$. We find a new kind of sufficient conditions, applicable to branching processes in a random environment. An important property of our estimates is that we don't necessary assume that ${\bf E}X_{i,1}^2$ are finite for every $i$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_05999 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Branching Process in a Varying Environment: How to Grow Like the Product of Means Kirpicheva, Y. Shklyaev, A. Probability 60J80 G.3 Consider a branching process $\{Z_n\}$ in a varying environment. Let $\{W_n\}$ be the natural martingale $Z_n/{\bf E}Z_n$. It converges to some random variable $W$ as $n\to\infty$. An important problem is to show that ${\bf P}(W>0)$ equals the survival probability, so that $Z_n$ is either $0$ or of the order ${\bf E}Z_n$. We find a new kind of sufficient conditions, applicable to branching processes in a random environment. An important property of our estimates is that we don't necessary assume that ${\bf E}X_{i,1}^2$ are finite for every $i$. |
| title | Branching Process in a Varying Environment: How to Grow Like the Product of Means |
| topic | Probability 60J80 G.3 |
| url | https://arxiv.org/abs/2604.05999 |