О локальном законе Марченко–Пастура для прореженных выборочных ковариационных матриц

The present talk is based on the results recently obtaibed jointly with F. Götze and D. Timushev.
Let $m=m(n)$, $m\ge n$. Consider independent identically distributed zero-mean random variables $X_{jk}$, $1\le j\le n$, $1\le k\le m$ with $\mathbb E X_{jk}=0$ , $\mathbb E X^2_{jk}=1$ and independent of that set independent Bernoulli random variables $\xi_{jk}$, $1\le j\le n$, $1\le k\le m$ with $\mathbb E\xi_{jk}=p_n$.
Consider the sequence of random matrices
$$ \mathbf X=\frac1{\sqrt{mp_n}}(\xi_{jk}X_{jk})_{1\le j\le n, 1\le k\le m}. $$
Denote by $s_1\ge \cdots\ge s_n$ the singular values of $\mathbf X$ and define the symmetrized empirical spectral distribution function (ESD) of the sample covariance matrix $\mathbf W=\mathbf X\mathbf X^*$:
$$ F_n(x)=\frac1{2n}\sum_{j=1}^n\Big(\mathbb I\{s_j\le x\}+\mathbb I\{-s_j\le x\}\Big), $$
where $\mathbb I\{A\}$ stands for the event $A$ indicator.
Let $y=y(n)=\frac nm$ and $G_y(x)$ — the symmetrized Marchenko–Pastur distribution function with the density
$$ g_y(x)=\frac 1{2\pi yx}\sqrt{(x^2-a^2)(b^2-x^2)}\,\mathbb I\{a^2\le x^2\le b^2\}, $$
where $a=1-\sqrt y,\quad b=1+\sqrt y$.
We investigate the asymptotics of Stieltjes transform of $F_n(x)$ in so called local regime under conditions

• for some $\delta>0$
  $$ \mathbb E|X_{jk}|^{4+\delta}=\mu_{4+\delta}<\infty; $$
  
• for $\varkappa=\frac{\delta}{2(4+\delta)}$
  $$ np_n\ge C\log^{\frac2{\varkappa}}n; $$
  
  
• 
  $$ |X_{jk}|\le C(np)^{\frac12-\varkappa}. $$