Smallest singular value of sparse random matrices

We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances from below. This allows us to consider matrices with null entries or, more generally, with entries having small variances. Our results do not assume identical distribution of the entries of a random matrix and help to clarify the role of the variances of the entries. We also show that it is enough to require boundedness from above of the $r$-th moment, $r > 2$, of the corresponding entries.Comment: 25 pages, a condition on one parameter was added in the statement of Theorem 1.3 and Lemma 6.2, results unchange

Forecasting and prequential validation for time varying meta-elliptical distributions

We consider forecasting and prequential (predictive sequential) validation of meta-elliptical distributions with time varying parameters. Using the weak prequential principle of Dawid, we conduct model validation avoiding nuisance parameter problems. Results rely on the structure of meta-elliptical distributions and we allow for discontinuities in the marginals and time varying parameters. We illustrate the ideas of the paper using a large data set of 16 commodity prices