  Using a set of evolution equations [J.G. Amar {\it et al}, Phys. Rev. Lett.
{\bf 86}, 3092 (2001)] for the average gap-size between islands, we calculate
analytically the asymptotic scaled capture-number distribution (CND) for
one-dimensional irreversible submonolayer growth of point islands. The
predicted asymptotic CND is in reasonably good agreement with kinetic
Monte-Carlo (KMC) results and leads to a \textit{non-divergent asymptotic}
scaled island-size distribution (ISD). We then show that a slight modification
of our analytical form leads to an analytic expression for the asymptotic CND
and a resulting asymptotic ISD which are in excellent agreement with KMC
simulations. We also show that in the asymptotic limit the self-averaging
property of the capture zones holds exactly while the asymptotic scaled gap
distribution is equal to the scaled CND.
