Understanding heavy tails in a bounded world or, is a truncated heavy tail heavy or not?

We address the important question of the extent to which random variables and vectors with truncated power tails retain the characteristic features of random variables and vectors with power tails. We define two truncation regimes, soft truncation regime and hard truncation regime, and show that, in the soft truncation regime, truncated power tails behave, in important respects, as if no truncation took place. On the other hand, in the hard truncation regime much of "heavy tailedness" is lost. We show how to estimate consistently the tail exponent when the tails are truncated, and suggest statistical tests to decide on whether the truncation is soft or hard. Finally, we apply our methods to two recent data sets arising from computer networks

Asymptotic behaviour of Gaussian minima

We investigate what happens when an entire sample path of a smooth Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high level is exceeded, the conditional shape of the process above the high level, and the location of the minimum of the process given that the sample path is above a high level.Chakrabarty's research was partially supported by the INSPIRE grant of the Department of Science and Technology, Government of India. Samorodnitsky's research was partially supported by the ARO grant W911NF-12-10385 and by the NSF grant DMS-1506783 at Cornell University

From random matrices to long range dependence

Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this work. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process, a phenomenon resembling that in the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behavior of the eigenvalues in a completely different way. Therefore, this helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.Comment: 50 pages. The current article generalises the results in http://arxiv.org/abs/1304.3394 and gives a new perspective and elegant proofs of some of the results there. It is to appear in Random Matrices: Theory and Application

Selection of Dominant Characteristic Modes

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.The theory of characteristic modes is a popular physics based deterministic approach which has found several recent applications in the fields of radiator design, electromagnetic interference modelling and radiated emission analysis. The modal theory is based on the approximation of the total induced current in an electromagnetic structure in terms of a weighted sum of multiple characteristic current modes. The resultant outgoing field is also a weighted summation of the characteristic field patterns. Henceforth, a proper modal measure is an essential requirement to identify the modes which play a dominant role for a frequency of interest. The existing literature of significance measures restricts itself for ideal lossless structures only. This paper explores the pros and cons of the existing measures and correspondingly suggests suitable alternatives for both radiating and scattering applications. An example is presented in order to illustrate the proposed modal method for approximating the shielding response of a slotted geometry