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Non-homogeneous renewal processes are not yet well established. One of thetools necessary for studying these processes is the non-homogeneous timeconvolution. Renewal theory has great relevance in general in economics and inparticular in actuarial science, however most actuarial problems are connectedwith the age of the insured person. The introduction of non-homogeneity in therenewal processes brings actuarial applications closer to the real world. Thispaper will define the non-homogeneous time convolutions and try to give orderto the non-homogeneous renewal processes. The numerical aspects of theseprocesses are dealt with and, finally, a real data application to an aspect ofmotorcar insurance is proposed.

Non-homogeneous time convolutions, renewal processes and age dependent mean number of motorcar accidents