  We take a wavelet based approach to the analysis of point processes and the
estimation of the first order intensity under a continuous time setting. A
multiresolution analysis of a point process is formulated which motivates the
definition of homogeneity at different scales of resolution, termed $J$-th
level homogeneity. Further to this, the activity in a point processes' first
order behavior at different scales of resolution is also defined and termed
$L$-th level innovation. Likelihood ratio tests for both these properties are
proposed with asymptotic distributions provided, even when only a single
realization of the point process is observed. The test for $L$-th level
innovation forms the basis for a collection of statistical strategies for
thresholding coefficients in a wavelet based estimator of the intensity
function. These thresholding strategies are shown to outperform the existing
local hard thresholding strategy on a range of simulation scenarios.
