Abstract: There has recently been a surge of interest in the statistics community for high-dimensional problems, i.e problems in multivariate statistics where the number of variables p is at least of the same order of magnitude as the number of observations, n. In this "large n, large p" setting, many "standard" estimators behave in perhaps unexpected ways, for instance exhibiting bias when their low-dimensional (p fixed) counterparts do not.

In this talk, I will discuss the classical Markowitz portfolio optimization problem, and show that its empirical solution differs quite significantly from its theoretical solution when the numbers of assets in the portfolio (p) is of the same order of magnitude as the number of observations (n) one uses to estimate the parameters of the problem.

Time permitting, I will also discuss briefly how some of the ideas used for gaining insights in the high- dimensional Markowitz problem can be used for the statistical question of high-dimensional sparse inverse covariance estimation.

Part of this work is joint with N.El Karoui and N. Touzi.