How close is the sample covariance matrix to the actual covariance matrix?

Details How close is the sample covariance matrix to the actual covariance matrix? How close is the sample covariance matrix to the actual covariance matrix?

2010-04-20

arxiv.org/abs/1004.3484v2

Mathematics

Probability

34 pages. Typos and minor inaccuracies corrected, references updated

Scientific paper

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.

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