Seminars: A. N. Tikhomirov, Об оценке преобразования Стилтьеса спектральной меры вблизи вещественной оси и оценки скорости сходимости в полукруговом законе и законе Марченко--Пастура

SEMINARS


Seminar on Probability Theory and Mathematical Statistics November 21, 2011 11:00, St. Petersburg, PDMI, room 311 (nab. r. Fontanki, 27)

Об оценке преобразования Стилтьеса спектральной меры вблизи вещественной оси и оценки скорости сходимости в полукруговом законе и законе Марченко–Пастура A. N. Tikhomirov
Abstract: Let $\mathbf X=(X_{jk})$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\le j\le k$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\mathbf X$ to the semi-circular law assuming that $\mathbb E X_{jk}=0$, $\mathbb E X_{jk}^2=1$ and that the distributions of the matrix elements $X_{jk}$ have a uniform sub exponential decay in the sense that there exists a constant $\varkappa>0$ such that for any $1\le j\le k\le n$ and any $t\ge 1$ we have $$ \Pr\{\|X_{jk}\|>t\}\le \varkappa^{-1}\exp\{-t^{\varkappa}\}. $$ By means of a short recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\mathbf W=\frac1{\sqrt n}\mathbf X$ and the semicircular law is of order $O(n^{-1}\log^b n)$ with some positive constant $b>0$. Similar result is obtaned for sample covariance matrices.