  In this work we prove an asymptotic result, that under some conditions on the
involved distribution functions, is valid for any Oppenheim expansion,
extending a classical result proven by W. Vervaat in 1972 for denominators of
the Luroth case. Furthermore, we study the convergence in distribution of
weighted sums of a sequence of independent random variables. Although the
result is of its own interest, in the present setting it is used to prove
convergence in distribution of specific sequences of random variables
generalizing known results obtained for Luroth random variables.
