  We consider the one-dimensional generalized forest fire process: at each site
of $\zz$, seeds and matches fall according some i.i.d. stationary renewal
processes. When a seed falls on an empty site, a tree grows immediately. When a
match falls on an occupied site, a fire starts and destroys immediately the
corresponding connected component of occupied sites. Under some quite
reasonable assumptions on the renewal processes, we show that when matches
become less and less frequent, the process converges, with a correct
normalization, to a limit forest fire model. According to the nature of the
renewal processes governing seeds, there are four possible limit forest fire
models. The four limit processes can be perfectly simulated. This study
generalizes consequently a previous result of the authors where seeds and
matches were assumed to fall according to Poisson processes.

Probability