Central limit theorems for random polytopes in a smooth convex set

Let $K$ be a smooth convex set with volume one in \BBR^d. Choose $n$ random points in $K$ independently according to the uniform distribution. The convex hull of these points, denoted by $K_n$, is called a {\it random polytope}. We prove that several key functionals of $K_n$ satisfy the central limit theorem as $n$ tends to infinity.Comment: 23 pages, no figur

Private manufacturing SMEs survival and growth in Vietnam: The role of export participation

This study investigates for the first time a linkage between export participation and firm performance in terms of survival and profit growth in Vietnam. Using an unbalanced panel dataset from 2005 to 2009, our study shows no difference in the survival probability between exporters and non-exporters. By digging deeper to export status at different stages, the results indicate that continuous exporters have a positive association with probability of survival whereas export stoppers indicate a negative relationship. In terms of the relationship between firm growth and export activity, using Average Treatment Effects (OLS), export status is not related to firm profit growth. However, the Quantile Treatment Effects estimates reveal that export participation is positively and statistically significant associated with firms having profit growth above the median. The above findings might imply that exporting promoting policies, coupled with policies maintaining positions of firms in export market could be helpful since this may help firms improve their survival probability and profit growth

Random matrices: Universal properties of eigenvectors

The four moment theorem asserts, roughly speaking, that the joint distribution of a small number of eigenvalues of a Wigner random matrix (when measured at the scale of the mean eigenvalue spacing) depends only on the first four moments of the entries of the matrix. In this paper, we extend the four moment theorem to also cover the coefficients of the \emph{eigenvectors} of a Wigner random matrix. A similar result (with different hypotheses) has been proved recently by Knowles and Yin, using a different method. As an application, we prove some central limit theorems for these eigenvectors. In another application, we prove a universality result for the resolvent, up to the real axis. This implies universality of the inverse matrix.Comment: 25 pages, no figures, to appear, Random Matrices: Theory and applications. This is the final version, incorporating the referee's suggestion

The spectrum of random kernel matrices: universality results for rough and varying kernels

We consider random matrices whose entries are f() or f(||Xi-Xj||^2) for iid vectors Xi in R^p with normalized distribution. Assuming that f is sufficiently smooth and the distribution of Xi's is sufficiently nice, El Karoui [17] showed that the spectral distributions of these matrices behave as if f is linear in the Marchenko--Pastur limit. When Xi's are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng and Singer [13]. Two results are shown in this paper: first it is shown that for a large class of distributions the regularity assumptions on f in El Karoui's results can be reduced to minimal; and secondly it is shown that the Gaussian assumptions in Cheng--Singer's result can be removed, answering a question posed in [13] about the universality of the limiting spectral distribution.Comment: 25 pages, referees' suggestions and corrections incorporated, to appear in Random Matrices: Theory and Application