General Exact Solution to the Problem of the Probability Density for Sums of Random Variables

The exact explicit expression for the probability density $p_N(x)$ for a sum of $N$ random, arbitrary correlated summands is obtained. The expression is valid for any number $N$ and any distribution of the random summands. Most attention is paid to application of the developed approach to the case of independent and identically distributed summands. The obtained results reproduce all known exact solutions valid for the, so called, stable distributions of the summands. It is also shown that if the distribution is not stable, the profile of $p_N(x)$ may be divided into three parts, namely a core (small $x$), a tail (large $x$), and a crossover from the core to the tail (moderate $x$). The quantitative description of all three parts as well as that for the entire profile is obtained. A number of particular examples are considered in detail.

Figure 1

Contour of integration for Eq. (11).

Figure 2

Plots of the Gauss law Eq. (2) shown as a thin full line and probability densities $p_N(z)$ calculated numerically according to Eqs. (6, 7, 12) at $l=2$ for $N={10}^2$ (thick dashed) and $N={10}^4$ (thick dot-dashed). Both $p_N(z)$ exhibit tails heavier than the Gaussian. The corresponding tail asymptotics given by Eq. (13) are drawn as thin dashed and dot-dashed lines, respectively.

Figure 3

Plots of probability density $p_N(z)$ calculated numerically according to Eqs. (6, 7) at $f(\xi )=1/(\pi \cosh \xi )$ and $N=25$ (thick line), the Gauss law Eq. (2) for the normalized sum $z$ (thin line) and asymptotics Eq. (14) shown as a dashed line. See the text for details.

Figure 4

Plots of the Gauss law Eq. (2) for the normalized sum $z$ (thin line) and probability density $p_N(z)$ calculated numerically according to Eqs. (6, 7, 16) at $N=10$ (thick line) exhibiting a tail lighter than the Gaussian. Note that deviation of $p_N(z)$ from the GL begins at $z\approx \sqrt{N}$.