It is well known that a class of subordinators can be
represented using the local time of Brownian motions. An
extension of such a representation is given for a class of
Lévy processes which are not necessarily of bounded
variation. This class can be characterized by the complete
monotonicity of the Lévy measures. The asymptotic behavior of
such processes is also discussed and the results are applied
to the generalized arc-sine law, an occupation time problem on
the positive side for one-dimensional diffusion processes.