Ergodic behavior of products of random positive operators
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929699327836160 |
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| author | Ligonnière, Maxime |
| author_facet | Ligonnière, Maxime |
| contents | This article is devoted to the study of products of random operators of the form $M_{0,n}=M_0\cdots M_{n-1}$, where $(M_{n})_{n\in\mathbb{N}}$ is an ergodic sequence of positive operators on the space of signed measures on a space $\mathbb{X}$. Under suitable conditions, in particular, a Doeblin-type minoration suited for non conservative operators, we obtain asymptotic results of the form \[ μM_{0,n} \simeq μ(\tilde{h}) r_n π_n,\] where $\tilde{h}$ is a random bounded function, $(r_n)_{n\geq 0}$ is a random non negative sequence and $π_n$ is a random probability measure on $\mathbb{X}$. Moreover, $\tilde{h}$, $(r_n)$ and $π_n$ do not depend on the choice of the measure $μ$. We prove additionally that $n^{-1} \log (r_n)$ converges almost surely to the Lyapunov exponent $λ$ of the process $(M_{0,n})_{n\geq 0}$ and that the sequence of random probability measures $(π_n)$ converges weakly towards a random probability measure. These results are analogous to previous estimates from Hennion in the case of $d\times d$ matrices, that were obtained with different techniques, based on a projective contraction in Hilbert distance. In the case where the sequence $(M_n)$ is i.i.d, we additionally exhibit an expression of the Lyapunov exponent $λ$ as an integral with respect to the weak limit of the sequence of random probability measures $(π_n)$ and exhibit an oscillation behavior of $r_n$ when $λ=0$. We provide a detailed comparison of our assumptions with the ones of Hennion and present some example of applications of our results, in particular in the field of population dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12088 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ergodic behavior of products of random positive operators Ligonnière, Maxime Probability This article is devoted to the study of products of random operators of the form $M_{0,n}=M_0\cdots M_{n-1}$, where $(M_{n})_{n\in\mathbb{N}}$ is an ergodic sequence of positive operators on the space of signed measures on a space $\mathbb{X}$. Under suitable conditions, in particular, a Doeblin-type minoration suited for non conservative operators, we obtain asymptotic results of the form \[ μM_{0,n} \simeq μ(\tilde{h}) r_n π_n,\] where $\tilde{h}$ is a random bounded function, $(r_n)_{n\geq 0}$ is a random non negative sequence and $π_n$ is a random probability measure on $\mathbb{X}$. Moreover, $\tilde{h}$, $(r_n)$ and $π_n$ do not depend on the choice of the measure $μ$. We prove additionally that $n^{-1} \log (r_n)$ converges almost surely to the Lyapunov exponent $λ$ of the process $(M_{0,n})_{n\geq 0}$ and that the sequence of random probability measures $(π_n)$ converges weakly towards a random probability measure. These results are analogous to previous estimates from Hennion in the case of $d\times d$ matrices, that were obtained with different techniques, based on a projective contraction in Hilbert distance. In the case where the sequence $(M_n)$ is i.i.d, we additionally exhibit an expression of the Lyapunov exponent $λ$ as an integral with respect to the weak limit of the sequence of random probability measures $(π_n)$ and exhibit an oscillation behavior of $r_n$ when $λ=0$. We provide a detailed comparison of our assumptions with the ones of Hennion and present some example of applications of our results, in particular in the field of population dynamics. |
| title | Ergodic behavior of products of random positive operators |
| topic | Probability |
| url | https://arxiv.org/abs/2312.12088 |