Conditioned local limit theorems for products of positive random matrices
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866911048664088576 |
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| author | Grama, Ion Xiao, Hui |
| author_facet | Grama, Ion Xiao, Hui |
| contents | Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices, where $d\geq 2$ is an integer. For any starting point $x \in \mathbb{R}_+^d$ with $|x| = 1$ and $y \in \mathbb R$, we define the exit time $τ_{x, y} = \inf \{ k \geq 1: y + \log |g_k \cdots g_1 x| < 0 \}$. In this paper, we investigate the conditioned local probability $\mathbb{P} (y + \log |g_n \cdots g_1 x| \in z + [0, Δ], τ_{x, y} > n)$ under various assumptions on $y$, $z$ and $Δ$. For the case where $z = O(\sqrt{n})$, we establish an exact asymptotic result as $n \to \infty$, uniformly in $y$ and $Δ$, which extends the classical Caravenna conditioned local limit theorem to the case of products of positive random matrices. Our proof does not rely on the reversibility techniques. Furthermore, for arbitrary $z \in \mathbb R_+$, we deduce a uniform upper bound with rate $n^{-3/2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07565 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Conditioned local limit theorems for products of positive random matrices Grama, Ion Xiao, Hui Probability Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices, where $d\geq 2$ is an integer. For any starting point $x \in \mathbb{R}_+^d$ with $|x| = 1$ and $y \in \mathbb R$, we define the exit time $τ_{x, y} = \inf \{ k \geq 1: y + \log |g_k \cdots g_1 x| < 0 \}$. In this paper, we investigate the conditioned local probability $\mathbb{P} (y + \log |g_n \cdots g_1 x| \in z + [0, Δ], τ_{x, y} > n)$ under various assumptions on $y$, $z$ and $Δ$. For the case where $z = O(\sqrt{n})$, we establish an exact asymptotic result as $n \to \infty$, uniformly in $y$ and $Δ$, which extends the classical Caravenna conditioned local limit theorem to the case of products of positive random matrices. Our proof does not rely on the reversibility techniques. Furthermore, for arbitrary $z \in \mathbb R_+$, we deduce a uniform upper bound with rate $n^{-3/2}$. |
| title | Conditioned local limit theorems for products of positive random matrices |
| topic | Probability |
| url | https://arxiv.org/abs/2310.07565 |