标题： 在最优矩假设下的独立随机矩阵强不可约乘积的极限定理 显示英文标题

标题： Limit theorems for a strongly irreducible product of independent random matrices under optimal moment assumptions

摘要： 设$ \nu $是在线性半群$ \mathrm{End}(E) $上的概率分布，对于$ E $是一个局部紧致域上的有限维向量空间。我们假设$ \nu $是接近的，强不可约的，并且$ \nu^{*n}\{0\}=0 $对所有整数$ n\in\mathbb{N} $成立。 我们考虑随机序列$ \overline\gamma_n := \gamma_0 \cdots \gamma_{n-1} $对于$ (\gamma_k)_{k \ge 0} $的独立分布律$ \nu $。 我们将对数奇异值间隙定义为$ \mathrm{sqz} = \log\left( \frac{\mu_1}{\mu_2} \right) $，其中$ \mu_1 $和$ \mu_2 $是两个最大的奇异值。 我们证明$ (\mathrm{sqz}(\overline{\gamma}_n))_{n\in\mathbb{N}} $线性地逃逸到无穷，并在其逃逸率以下满足指数型大偏差估计。 在相同假设下，我们还证明了由$ \overline{\gamma}_n $映射的任意一条直线以及其最大特征值的特征空间都以指数速度收敛到同一条随机直线$l_\infty $。如果我们进一步假设推送前分布$N(\nu)$对于$ N:g\mapsto\log\left(\|g\|\|g^{-1}\|\right) $和某个$ p\ge 1 $是$ \mathrm{L}^p $，则我们证明了对于所有单位线性形式$ w $，$ \log|w(l_\infty)| $是$ \mathrm{L}^p $，并且$ \overline{\gamma}_n $的每个系数的对数几乎必然等价于范数的对数。 为了证明这些结果，我们不依赖于任何关于可逆矩阵随机乘积的经典结果，这些结果需要$ \mathrm{L}^1 $的矩假设。 相反，我们描述了一种有效的方法，将独立同分布的因子分组为在Cartan投影中对齐的独立同分布随机词。 此外，我们对矩有明确的控制。 显示英文摘要

摘要： Let $ \nu $ be a probability distribution over the linear semi-group $ \mathrm{End}(E) $ for $ E $ a finite dimensional vector space over a locally compact field. We assume that $ \nu $ is proximal, strongly irreducible and that $ \nu^{*n}\{0\}=0 $ for all integers $ n\in\mathbb{N} $. We consider the random sequence $ \overline\gamma_n := \gamma_0 \cdots \gamma_{n-1} $ for $ (\gamma_k)_{k \ge 0} $ independents of distribution law $ \nu $. We define the logarithmic singular gap as $ \mathrm{sqz} = \log\left( \frac{\mu_1}{\mu_2} \right) $ , where $ \mu_1 $ and $ \mu_2 $ are the two largest singular values. We show that $ (\mathrm{sqz}(\overline{\gamma}_n))_{n\in\mathbb{N}} $ escapes to infinity linearly and satisfies exponential large deviations estimates below its escape rate. With the same assumptions, we also show that the image of a generic line by $ \overline{\gamma}_n $ as well as its eigenspace of maximal eigenvalue both converge to the same random line $l_\infty $ at an exponential speed.If we moreover assume that the push-forward distribution $N(\nu)$ is $ \mathrm{L}^p $ for $ N:g\mapsto\log\left(\|g\|\|g^{-1}\|\right) $ and for some $ p\ge 1 $, then we show that $ \log|w(l_\infty)| $ is $ \mathrm{L}^p $ for all unitary linear form $ w $ and the logarithm of each coefficient of $ \overline{\gamma}_n $ is almost surely equivalent to the logarithm of the norm. To prove these results, we do not rely on any classical results for random products of invertible matrices with $ \mathrm{L}^1 $ moment assumption. Instead we describe an effective way to group the i.i.d factors into i.i.d random words that are aligned in the Cartan projection. We moreover have an explicit control over the moments.