Asymptotic distribution of number of coincidences of a bivariate sample under natural matching (Q1603030)

Let \(((t_1,\dots, t_n)\), \((u_1,\dots, u_n))\) be a sample from a bivariate population which was componentwise extracted so that it is unknown which \(u_i\) value correponds to which \(t_j\) value. The two order statistics are used to restore the original pairs. Let \(N\) be the number of coincidences with the original pairs. The main result is an integral representation of all moments of \(N\) which gives the fact that the asymptotic distribution of \(N\) is a Poisson distribution, as \(n\to\infty\).