  We study the asymptotics of the moments of arithmetic functions that have a
limit distribution, not necessarily normal, defined on a subset of the natural
series that satisfies certain requirements. Several assertions are proved on
estimating the asymptotics of the moments of strongly additive arithmetic
functions and also with additive functions of the class H that have a limit
distribution and are defined on a subset of the natural series. The first
version of the article is devoted to the study of the asymptotics of the
moments of arithmetic functions that have a limit distribution on the natural
series. The second version of the article is devoted to the study of the
asymptotics of the moments of arithmetic functions that have a limit
distribution on an arithmetic progression.
