Tails of probability density for sums of random independent variables

The exact expression for the probability density $p_{_N}(x)$ for sums of a finite number $N$ of random independent terms is obtained. It is shown that the very tail of $p_{_N}(x)$ has a Gaussian form if and only if all the random terms are distributed according to the Gauss Law. In all other cases the tail for $p_{_N}(x)$ differs from the Gaussian. If the variances of random terms diverge the non-Gaussian tail is related to a Levy distribution for $p_{_N}(x)$. However, the tail is not Gaussian even if the variances are finite. In the latter case $p_{_N}(x)$ has two different asymptotics. At small and moderate values of $x$ the distribution is Gaussian. At large $x$ the non-Gaussian tail arises. The crossover between the two asymptotics occurs at $x$ proportional to $N$. For this reason the non-Gaussian tail exists at finite $N$ only. In the limit $N$ tends to infinity the origin of the tail is shifted to infinity, i. e., the tail vanishes. Depending on the particular type of the distribution of the random terms the non-Gaussian tail may decay either slower than the Gaussian, or faster than it. A number of particular examples is discussed in detail.Comment: 6 pages, 4 figure

Off-resonance field enhancement by spherical nanoshells

We study light scattering by spherical nanoshells consistent of metal/dielectric composites. We consider two geometries of metallic nanoshell with dielectric core, and dielectric coated metallic nanoparticle. We demonstrate that for both geometries the local field enhancement takes place out of resonance regions ("dark states"), which, nevertheless, can be understood in terms of the Fano resonance. At optimal conditions the light is stronger enhanced inside the dielectric material. By using nonlinear dielectric materials it will lead to a variety nonlinear phenomena applicable for photonics applications