Limiting curves for i.i.d. records (Q1897165)

For independent bivariate vectors \((X_i, Y_i)\), \(i \geq 1\), define \(L_n\) as \(L_n = \infty\) if \(L_{n - 1} = \infty\) and \(L_n = \inf \{m > L_{n - 1} : X_m > \max (X_j : 1 \leq j \leq m - 1)\) and \(Y_m > \max (Y_j : 1 \leq j \leq m - 1)\}\) if \(L_{n - 1} < \infty\). Set \(R_n = (X_{L_n}, Y_{L_n})\) if \(L_n < \infty\). The authors show that, under some assumptions, the records \(R_n\) concentrate around limiting curves which form the solution of a variational problem. This variational problem is studied in detail. The relation of the preceding problem to the longest increasing subsequence problem is pointed out.