Positron scattering from the vibrational ground state of the hydrogen molecular ion has been investigated using the single-centre convergent close-coupling (CCC) method. Cross sections for dissociative excitation, dissociative ionisation, total inelastic scattering, and proton production have been calculated at incident positron energies from 10 to 500 eV. For this energy range the mass stopping power and mean excitation energy have also been calculated for both the positron and electron cases. Since there are no experiments or other calculations for positron scattering on the hydrogen molecular ion, the obtained results have been compared with corresponding CCC results for positron collisions with He+ and electron collisions with
. Comparative analysis allows to estimate the quality of the approximations used in the present calculations.