Tails of probability density for sums of random independent variables

摘要

The exact expression for the probability density $p_{_N}(x)$ for sums of afinite number $N$ of random independent terms is obtained. It is shown that thevery tail of $p_{_N}(x)$ has a Gaussian form if and only if all the randomterms are distributed according to the Gauss Law. In all other cases the tailfor $p_{_N}(x)$ differs from the Gaussian. If the variances of random termsdiverge the non-Gaussian tail is related to a Levy distribution for$p_{_N}(x)$. However, the tail is not Gaussian even if the variances arefinite. In the latter case $p_{_N}(x)$ has two different asymptotics. At smalland moderate values of $x$ the distribution is Gaussian. At large $x$ thenon-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 shiftedto infinity, i. e., the tail vanishes. Depending on the particular type of thedistribution of the random terms the non-Gaussian tail may decay either slowerthan the Gaussian, or faster than it. A number of particular examples isdiscussed in detail.