  Non-homogeneous renewal processes are not yet well established. One of the
tools necessary for studying these processes is the non-homogeneous time
convolution. Renewal theory has great relevance in general in economics and in
particular in actuarial science, however most actuarial problems are connected
with the age of the insured person. The introduction of non-homogeneity in the
renewal processes brings actuarial applications closer to the real world. This
paper will define the non-homogeneous time convolutions and try to give order
to the non-homogeneous renewal processes. The numerical aspects of these
processes are dealt with and, finally, a real data application to an aspect of
motorcar insurance is proposed.
