Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How close is the sample covariance matrix to the actual covariance
  matrix?   Given a probability distribution in R^n with general (non-white) covariance,
a classical estimator of the covariance matrix is the sample covariance matrix
obtained from a sample of N independent points. What is the optimal sample size
N = N(n) that guarantees estimation with a fixed accuracy in the operator norm?
Suppose the distribution is supported in a centered Euclidean ball of radius
\sqrt{n}. We conjecture that the optimal sample size is N = O(n) for all
distributions with finite fourth moment, and we prove this up to an iterated
logarithmic factor. This problem is motivated by the optimal theorem of
Rudelson which states that N = O(n \log n) for distributions with finite second
moment, and a recent result of Adamczak, Litvak, Pajor and Tomczak-Jaegermann
which guarantees that N = O(n) for sub-exponential distributions.

8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys