Georgia Institute of Technology College of Sciences

We provide a sufficient condition for the continuity of real valued permanental processes. When applied to the subclass of permanental processes which consists of squares of Gaussian processes, we obtain the sufficient condition for continuity which is also known to be necessary. Using an isomorphism theorem of Eisenbaum and Kaspi which relates Markov local times and permanental processes we obtain a general sufficient condition for the joint continuity of the local times.

In this talk we shall discuss the problem of fitting a distribution function to the marginal distribution of a long memory process. It is observed that unlike in the i.i.d. set up, classical tests based on empirical process are relatively easy to implement. More importantly, we discuss fitting the marginal distribution of the error process in location, scale and linear regression models. An interesting observation is that the first order difference between the residual empirical process and the null model can not be used to asymptotically to distinguish between the two marginal distributions that differ only in their means. This finding is in sharp contrast to a recent claim of Chan and Ling to appear in the Ann. Statist. that such a process has a Gaussian weak limit. We shall also proposes some tests based on the second order difference in this case and analyze some of their properties. Another interesting finding is that residual empirical process tests in the scale problem are robust against not knowing the scale parameter. The third finding is that in linear regression models with a non-zero intercept parameter the first order difference between the empirical d.f. of residuals and the null d.f. can not be used to fit an error d.f. This talk is based on ongoing joint work with Donatas Surgailis.

The almost sure rate of convergence in the sup norm for linear wavelet density estimators is obtained, as well as a central limit theorem for the distribution functions based on these estimators. These results are then applied to show that the hard thresholding wavelet estimator of Donoho, Johnstone, Kerkyacharian and Picard (1995) is adaptive in sup norm to the smoothness of a density. An alternative adaptive estimator combining Lepski's method with Rademacher complexities will also be described. This is joint work with Richard Nickl.

The talk will present several limit theorems for the supercritical colony of the particles with masses. Reaction-diffusion equations responsible for the spatial distribution of the species contain the usual random death, birth and migration processes. The evolution of the mass of the individual particle includes (together with the diffusion) the mitosis: the splitting of the mass between the two offspring. The last process leads to the new effects. The limit theorems give the detailed picture of the space –mass distribution of the particles in the bulk of the moving front of the population.