  Collective flavor oscillations are known to bring multiple splits in the
supernova (SN) neutrino and antineutrino spectra. These spectral splits depend
not only on the mass hierarchy of the neutrinos but also on the initial
relative flux composition. Observation of spectral splits in a future galactic
supernova signal is expected to throw light on the mass hierarchy pattern of
the neutrinos. However, since the Diffuse Supernova Neutrino Background (DSNB)
comprises of a superposition of neutrino fluxes from all past supernovae, and
since different supernovae are expected to have slightly different initial
fluxes, it is pertinent to check if the hierarchy dependent signature of
collective oscillations can survive this averaging of the flux spectra. Since
the actual distribution of SN with initial relative flux spectra of the
neutrinos and antineutrinos is unknown, we assume a log-normal distribution for
them. We study the dependence of the hierarchy sensitivity to the mean and
variance of the log-normal distribution function. We find that the hierarchy
sensitivity depends crucially on the mean value of the relative initial
luminosity. The effect of the width is to reduce the hierarchy sensitivity for
all values of the mean initial relative luminosity. We find that in the very
small mixing angle ($\theta_{13}$) limit considering only statistical errors
even for very moderate values of variance, there is almost no detectable
hierarchy sensitivity if the mean relative luminosities of $\nu_e$ and
$\bar\nu_e$ are greater than 1.
