强不可约独立随机矩阵乘积的极限定理
Limit theorems for a strongly irreducible product of independent random matrices under optimal moment assumptions
AI总结：
本文研究了在最优矩假设下，强不可约独立随机矩阵乘积的极限行为，通过分析奇异值和Lyapunov指数，证明了某些量以线性速率逃离无穷远并满足大偏差不等式，扩展了顶部Lyapunov指数的简单性概念。
摘要：
设ν是一个在半群上的概率分布，该半群由大小为d≥2的方阵组成，例如实数域R。我们考虑随机游走\overlineγ_n := γ_0⋯γ_{n−1}，其中(γ_k)_{k∈N}独立服从分布ν。设s_1≥s_2≥⋯≥s_d为由Cartan投影给出的奇异值。在ν的收缩假设下，我们证明了(log(s_1/s_2)(\overlineγ_n))_{n∈N}以线性速率逃离无穷远并满足其逃逸速率下的指数大偏差不等式。这扩展了顶部Lyapunov指数的简单性概念。我们还证明了随机线通过\overlineγ_n的像及其最大特征值的特征空间都以指数速度收敛到同一个随机线ℓ^∞。如果进一步假设ν在可逆矩阵群上，并且推 forward 分布N_*ν是L^p对于N: g ↦ log‖g‖‖g^{-1}‖和某些p > 0，则我们证明−log d(ℓ^∞, H)对于所有适当子空间H⊂R^d是统一的L^p。对于p=1，我们还证明每个\overlineγ_n的系数的缩放对数几乎必然收敛到顶部Lyapunov指数。为了证明这些结果，我们不依赖于平稳测度或Lyapunov指数的存在性。相反，我们描述了一种有效的方法将i.i.d.因子分组为i.i.d.随机词，这些词在Cartan分解中以某种方式对齐。我们还对矩有显式控制。
英文摘要：
Let $ν$ be a probability distribution over the semi-group of square matrices of size $d \ge 2$ over a locally compact field $\mathbb{K}$, \textit{e.g.} $\mathbb{R}$. We consider the random walk $\overlineγ_n := γ_0\cdotsγ_{n-1}$ for $(γ_k)_{k \in \mathbb{N}}$ independent of law $ν$. Let $s_1 \ge s_2 \ge \dots \ge s_d$ be the singular values given by the Cartan projection. Under a contraction assumption on $ν$, we show that $(\log\frac{s_1}{s_2}(\overlineγ_n))_{n \in\mathbb{N}}$, escapes to infinity linearly and satisfies exponential large deviations inequalities below its escape rate. This extends the notion of simplicity of the top Lyapunov exponent. We also show that the image of a generic line by $\overlineγ_n$ as well as its eigenspace of maximal eigenvalue both converge to the same random line $\ell^\infty$ at an exponential speed. If we moreover assume that $ν$ is supported on the group of invertible matrices and that the push-forward distribution $N_*ν$ is $\mathrm{L}^p$ for $N: g \mapsto\log\|g\|\|g^{-1}\|$ and for some $p > 0$, then we show that $- \log\mathrm{d}(\ell^\infty, H)$ is uniformly $\mathrm{L}^p$ for all proper subspace $H \subset \mathbb{R}^d$. For $p = 1$, we moreover show that the rescaled logarithm of each coefficient of $\overlineγ_n$ almost surely converges to the top Lyapunov exponent. To prove these results, we do not rely on the existence of the stationary measure nor on the existence of the Lyapunov exponents. Instead we describe an effective way to group the i.i.d. factors into i.i.d. random words that are somehow aligned in the Cartan decomposition. We moreover have an explicit control over the moments.
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