  In a functional setting, we propose two test statistics to highlight the
Poisson nature of a Cox process when n copies of the process are available. Our
approach involves a comparison of the empirical mean and the empirical variance
of the functional data and can be seen as an extended version of a classical
overdispersion test for counting data. The limiting distributions of our
statistics are derived using a functional central limit theorem for c`adl`ag
martingales. We also study the asymptotic power of our tests under some local
alternatives. Our procedure is easily implementable and does not require any
knowledge of covariates. A numerical study reveals the good performances of the
method. We also present two applications of our tests to real data sets.
