On the Rate of Convergence to the Marchenko--Pastur Distribution

Details On the Rate of Convergence to the Marchenko--Pastur Distribution On the Rate of Convergence to the Marchenko--Pastur Distribution

2011-10-06

arxiv.org/abs/1110.1284v1

Mathematics

Probability

Scientific paper

Let $\bold X=(X_{jk})$ denote $n\times p$ random matrix with entries $X_{jk}$, which are independent for $1\le j\le n,1\le k\le p$. We consider the rate of convergence of empirical spectral distribution function of matrix $\bold W=\frac1p\bold X\bold X^*$ to the Marchenko--Pastur law. We assume that $\E X_{jk}=0$, $\E X_{jk}^2=1$ and that the distributions of the matrix elements $X_{jk}$ have a uniformly sub exponential decay in the sense that there exists a constant $\varkappa>0$ such that for any $1\le j \le n,\,1\le k\le p $ and any $t\ge 1$ we have $$ \Pr\{|X_{jk}|>t\}\le \varkappa^{-1}\exp\{-t^{\varkappa}\}. $$ By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the sample covariance matrix $\mathbf W$ and the Marchenko--Pastur distribution is of order $O(n^{-1}\log^b n)$ with some positive constant $b>0$.

Affiliated with

Also associated with

No associations

Rating

On the Rate of Convergence to the Marchenko--Pastur Distribution does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the Rate of Convergence to the Marchenko--Pastur Distribution, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Rate of Convergence to the Marchenko--Pastur Distribution will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-181724