With the operation of the worldwide new radioactive ion beam facilities and the developments in the detection techniques, entirely unexpected features have been found in the unstable neutron-rich nuclei far from the stability line, e.g., the neutron halo structure. The main characteristic of the halo is a long tail in the nuclear density distribution. In the single-particle picture, the asymptotic tail of the density is described in terms of the binding energy of the last bound single-particle orbit. This is true without the pair correlation between the nucleons. However, pair correlation is known to play an important role in the formation of the neutron halo structure. In this talk, we will discuss the explicit effect of the pair correlation on the asymptotic tails of the diffusive neutron density and pair density or the pairingtensor, which represents the condensate of the nucleon pair distribution, in the neutron rich nuclei. We would like to clarify what mechanism governs the asymptotic slope of these densities and what are their key compositions. We apply the Hartree-Fock-Bogoliubov (HFB) theory, which can properly describe the pairing properties in the neutron-rich nucleus. We will demonstrate that in the extremely weakly bound nucleus, the non-resonant continuum state, instead of the weakly bound or resonant single-particle states around the Fermi energy, will dominate the asymptotic tail of the neutron density due to the pair correlation. For the pair density, we find a ‘universal’ asymptotic tail, whose exponential slope is determined only by the Fermi energy. This slope holds for the nucleus no matter it is deeply or weakly bound. We also find that the high angular momentum partial waves are important compositions in the pair density. The slope and the high angular momentum composition of the pair density indicate that a spatially correlated Cooper pair, whose binding energy is the two-neutron separation energy, penetrates far outside the nucleus.