  We present a systematic study of the statistics of the occupation time and
related random variables for stochastic processes with independent intervals of
time. According to the nature of the distribution of time intervals, the
probability density functions of these random variables have very different
scalings in time. We analyze successively the cases where this distribution is
narrow, where it is broad with index $\theta <1$, and finally where it is broad
with index $1<\theta <2$. The methods introduced in this work provide a basis
for the investigation of the statistics of the occupation time of more complex
stochastic processes (see joint paper by G. De Smedt, C. Godr\`{e}che, and J.M.
Luck).
