On the CLT for Linear Eigenvalue Statistics of a Tensor Model of Sample Covariance Matrices

Автор(и)

  • Alicja Dembczak-Kołodziejczyk University of Opole, 48 Oleska, Opole, 45-052, Poland
  • Anna Lytova University of Opole, 48 Oleska, Opole, 45-052, Poland

DOI:

https://doi.org/10.15407/mag19.02.374

Анотація

В [18] було доведено центральну граничну теорему (ЦГТ) для лiнiйних статистик власних значень ${Tr} \varphi(M_n)$ вибiркових коварiацiйних матриць $M_{n}=\sum_{\alpha=1}^m {\bf y}_{\alpha}^{(1)} \otimes {\bf y}_{\alpha}^{(2)}({\bf y}_{\alpha}^{(1)} \otimes {\bf y}_{\alpha}^{(2)})^T$, де $({\bf y}_{\alpha}^{(1)},\, {\bf y}_{\alpha}^{(2)})_{\alpha}$ є незалежними копiями вектора ${\bf y}\in \mathbb{R}^n$ , що задовольняє умови ${\bf E} {\bf y}{\bf y}^T=n^{-1} I_n$, ${\bf E} {\bf y}^2_i{\bf y}^2_j(1+\delta_{ij}d)n^{-2}+a(1+\delta_{ij}d_1)n^{-3}+O(n^{-4})$ для деяких $a, d,d_1\in \mathbb{R}$. Було показано, що для достатньо гладких тестових функцiй $\varphi$ маємо ${\bf Var} {Tr} \varphi(M_n)=O(n)$, коли $m, n\to\infty $, $m/n^2\to c>0$, крiм того $({Tr} \varphi(M_n)-{\bf E} {Tr} \varphi(M_n))/\sqrt{n}$ збiгається за розподiлом до гаусiвської випадкової величини з нульовим середнiм та дисперсiєю $V[\varphi]$ пропорцiйною $a+d$. Зокрема, якщо ${\bf y}$ рiвномiрно розподiлено на одиничнiй сферi, то $a+d=0$ i $V[\varphi]$. У цiй роботi ми показуємо, що в цьому випадку ${\bf Var}  {Tr}(M_n-zI_n)^{-1}=O(1)$, так що ЦГТ має бути справедливою для самих лiнiйних статистик власних значень без нормалiзувального коефiцiєнта (на вiдмiну вiд випадку вiдповiдних гаусiвських вибiркових коварiацiйних матриць).

Mathematical Subject Classification 2020: 60B12, 60B20, 60F05, 47A75

Ключові слова:

вибiрковi коварiацiйнi матрицi, центральна гранична теорема, лiнiйна статистика власних значень

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Dembczak-Kołodziejczyk, A.; Lytova, A. On the CLT for Linear Eigenvalue Statistics of a Tensor Model of Sample Covariance Matrices. Журн. мат. фіз. анал. геом. 2023, 19, 374–395.

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