当p趋于无穷大时pxq-矩阵上径向随机游动的极限定理
Limit theorems for radial random walks on pxq-matrices as p tends to infinity
摘要：
$R^p$上的径向概率测度与$[0,\infty[$上的概率测度一一对应。欧氏范数映射对于固定的$ν\in M^1([0,\infty[)$和每个维度p，我们考虑i.i.d. $R^p$ -值随机变量$X_1^p,X_2^p,...$，具有与上述$ν$对应的径向定律。本文以适当的方法导出了欧氏长度过程$S_k^p:=\|X_1^p+...+X_k^p\|$为k，p \to \infty的弱大数定律和强大数定律以及大偏差原理.事实上，我们得到这些结果在一个更高的秩设置，其中$R^p$是由空间的$p\times q$矩阵和$[0,\infty[$由锥$Π_q$的半正定矩阵。证明是基于这样一个事实，即$(S_k^p)_{k\ge 0}$形式马尔可夫链上的锥的转移概率给出的贝塞尔函数$J_μ$的矩阵参数与指数$μ$取决于p.的极限定理遵循新的渐近结果为$J_μ$为$μ\to \infty$。类似的结果也证明了某些Dunkl型贝塞尔函数。
英文摘要：
The radial probability measures on $R^p$ are in a one-to-one correspondence with probability measures on $[0,\infty[$ by taking images of measures w.r.t. the Euclidean norm mapping. For fixed $ν\in M^1([0,\infty[)$ and each dimension p, we consider i.i.d. $R^p$-valued random variables $X_1^p,X_2^p,...$ with radial laws corresponding to $ν$ as above. We derive weak and strong laws of large numbers as well as a large deviation principle for the Euclidean length processes $S_k^p:=\|X_1^p+...+X_k^p\|$ as k,p\to\infty in suitable ways. In fact, we derive these results in a higher rank setting, where $R^p$ is replaced by the space of $p\times q$ matrices and $[0,\infty[$ by the cone $Π_q$ of positive semidefinite matrices. Proofs are based on the fact that the $(S_k^p)_{k\ge 0}$ form Markov chains on the cone whose transition probabilities are given in terms Bessel functions $J_μ$ of matrix argument with an index $μ$ depending on p. The limit theorems follow from new asymptotic results for the $J_μ$ as $μ\to \infty$. Similar results are also proven for certain Dunkl-type Bessel functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
