  We developed a general method for assessing the evolution of the energy
spectrum of relativistic charged particles that have undergone small quantum
losses, such as the ionization losses when the electrons pass through matter
and the radiation losses in the periodic fields. These processes are
characterized by a small magnitude of the recoil quantum as compared with the
particle's initial energy. We convey the statistical consideration of the
radiating electrons. We demonstrate that in the initial stage, the electron's
spectrum can be described as a composition of consecutive convolutions of the
recoil spectrum with itself, taken with the Poisson mass. In this stage, at the
relatively small average number of the recoils, the electron spectrum reveals
some individual characteristics of the recoil spectrum. Furthermore, the
spectrum loses individuality and allows for proximate description in the terms
of statistical parameters. The results of this consideration reveal that the
width of the electron spectrum is increasing with the number of recoils
according to the power law, with the power index being inverse to the stability
parameter, which gradually increases with the number of recoils from one to
two. Increase of the spectrum width limits the ability of the beam to generate
coherent radiation in hard x-ray and gamma-ray region.
