Research ArticlesNo Access

Central limit theorems for the real eigenvalues of large Gaussian random matrices

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

https://doi.org/10.1142/S2010326317500022Cited by:15 (Source: Crossref)

Abstract

Let $G$ $G$ be an $N \times N$ $N \times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{i j} \sim 𝒩 (0, 1)$ . The eigenvalues of such matrices are known to form a two-component system consisting of purely real and complex conjugated points. The purpose of this paper is to show that by appropriately adapting the methods of [E. Kanzieper, M. Poplavskyi, C. Timm, R. Tribe and O. Zaboronski, Annals of Applied Probability26(5) (2016) 2733–2753], we can prove a central limit theorem of the following form: if $λ_{1}, \dots, λ_{N_{ℝ}}$ are the real eigenvalues of $G$ , then for any even polynomial function $P (x)$ and even $N = 2 n$ , we have the convergence in distribution to a normal random variable

1√𝔼(Nℝ)(Nℝ∑j=1P(λj/√2n)−𝔼Nℝ∑j=1P(λj/√2n))→𝒩(0,σ2(P))<math display="block" altimg="eq-00008.gif"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mi>𝔼</mi><mo class="MathClass-open" stretchy="false">(</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>ℝ</mi></mrow></msub><mo class="MathClass-close" stretchy="false">)</mo></mrow></msqrt></mrow></mfrac><mfenced separators="" open="(" close=")"><mrow><munderover accentunder="true" accent="true"><mrow><mo class="MathClass-op">∑</mo></mrow><mrow><mi>j</mi><mo class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>ℝ</mi></mrow></msub></mrow></munderover><mi>P</mi><mo class="MathClass-open" stretchy="false">(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false" class="MathClass-bin">/</mo><msqrt><mrow><mn>2</mn><mi>n</mi></mrow></msqrt><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">−</mo><mi>𝔼</mi><munderover accentunder="true" accent="true"><mrow><mo class="MathClass-op">∑</mo></mrow><mrow><mi>j</mi><mo class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>ℝ</mi></mrow></msub></mrow></munderover><mi>P</mi><mo class="MathClass-open" stretchy="false">(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false" class="MathClass-bin">/</mo><msqrt><mrow><mn>2</mn><mi>n</mi></mrow></msqrt><mo class="MathClass-close" stretchy="false">)</mo></mrow></mfenced><mo class="MathClass-rel">→</mo><mi mathvariant="cal">𝒩</mi><mo class="MathClass-open" stretchy="false">(</mo><mn>0</mn><mo class="MathClass-punc">,</mo><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo class="MathClass-open" stretchy="false">(</mo><mi>P</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">)</mo></mrow></math>

$n \to \infty$ , where

$σ^{2} (P) = \frac{2 - \sqrt{2}}{2} \int_{- 1}^{1} P {(x)}^{2} d x$ .

Keywords:

AMSC: 60B20, 60F05, 15B52

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