Central Limit Theorems for Radial Random Walks on \(p\times q\) Matrices for \(p\to\infty\)

Let \(\nu\in M^1([0,\infty[)\) be a fixed probability measure. For each dimension \(p\in\b N\), let \((X_n^p)_{n\ge1}\) be i.i.d. \(\b R^p\)-valued radial random variables with radial distribution \(\nu\). We derive two central limit theorems for \( \|X_1^p+...+X_n^p\|_2\) for \(n,p\to\infty\) with...

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Bibliographic Details
Published inarXiv.org
Main Author Voit, Michael
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 18.01.2012
Subjects
Asymptotic properties
Radial distribution
Random variables
Random walk
Random walk theory
Theorems
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Summary:Let \(\nu\in M^1([0,\infty[)\) be a fixed probability measure. For each dimension \(p\in\b N\), let \((X_n^p)_{n\ge1}\) be i.i.d. \(\b R^p\)-valued radial random variables with radial distribution \(\nu\). We derive two central limit theorems for \( \|X_1^p+...+X_n^p\|_2\) for \(n,p\to\infty\) with normal limits. The first CLT for \(n>>p\) follows from known estimates of convergence in the CLT on \(\b R^p\), while the second CLT for \(n<<p\) will be a consequence of asymptotic properties of Bessel convolutions. Both limit theorems are considered also for \(U(p)\)-invariant random walks on the space of \(p\times q\) matrices instead of \(\b R^p\) for \(p\to\infty\) and fixed dimension \(q\).
ISSN:2331-8422