Thin-shell theory for rotationally invariant random simplices

Heiny, Johannes; Johnston, S. R.; Prochno, Joscha

doi:10.48550/arxiv.2103.11872

Cited by 1 publication

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References 48 publications

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“…matrices under existence of the 4th moments of matrix entries, to mention a few. In some special cases where a suitable stochastic representation is available, Grote et al (2019) also proved large deviation results and Heiny et al (2021a) fast Berry-Esseen bounds. Our particular interest covers the sample correlation matrix denoted by R which is computed from a random sample y 1 .…”

Section: Introductionmentioning

confidence: 87%

Logarithmic law of large random correlation matrix

Parolya¹,

Heiny²,

Kurowicka³

2021

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Consider a random vector y = Σ 1/2 x, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and Σ 1/2 is a deterministic p × p matrix such that the spectral norm of the population correlation matrix R of y is uniformly bounded. In this paper, we find that the log determinant of the sample correlation matrix R based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for p/n → γ ∈ (0, 1] and p ≤ n. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of y is unknown, we show that after recentering by the empirical mean the obtained CLT holds with a shift in the asymptotic mean. This result is of independent interest in both large dimensional random matrix theory and high-dimensional statistical literature of large sample correlation matrices for non-normal data. At last, the obtained findings are applied for testing of uncorrelatedness of p random variables. Surprisingly, in the null case R = I, the test statistic becomes completely pivotal and the extensive simulations show that the obtained CLT also holds if the moments of order four do not exist at all, which conjectures a promising and robust test statistic for heavy-tailed high-dimensional data.

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Section: Introductionmentioning

confidence: 87%