AM signal detection with Hilbert?

I need to perform detection and estimation on a signal which contains
amplitude-modulated pulses.  The model for the signal is

   y[n] = e[n] + sum(Ek[n-Tk])

where e[n] is noise, Ek[] is a pulse (product of hamming-esque window
and a sinusoid), and Tk the corresponding lag.  The problem is that the
envelope gain, carrier frequency, and event duration are all random
variables.  I can visually spot the events by looking at the amplitude
envelope function estimated from the Hilbert transform, but the
variable gain, duration, and frequency have stumped me on implementing
a detection algorithm!?

Can anyone offer thoughts, ideas, tips, references, etc. for this
problem?  I'm considering thresholding IF and IA, or sparse
deconvolution of IA, but both seem problematic.  Also, is there a name
for this type of signal in the communications/dsp literature?

Thanks,
Marcus