Short Course

With Professor Vladimir Koltchinskii, Georgia Tech.

Short Course: Lecture 1 – Sample Covariance Operators: Normal Approximation and Concentration

Short Course Lecture 1 –

In this short course, several problems related to statistical estimation of covariance operators

and their spectral characteristics will be discussed. The problems will be studied in a dimension-free framework in which the data lives in high-dimensional or infinite-dimensional spaces and “complexity”

of estimation is characterized by the so called “effective rank’’ of the true covariance operator rather than by the dimension of the ambient space. In this framework, sharp moment bounds and concentration inequalities for the operator norm error of sample covariance will be proved

in the Gaussian case showing that the “effective rank’’ characterizes the size of this error.

In addition to this, a number of recent results on normal approximation and concentration of

functions of sample covariance operators, including their spectral projections, will be discussed.

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