## Limiting curves for i.i.d. records.(English)Zbl 0834.60058

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.

### MSC:

 60G70 Extreme value theory; extremal stochastic processes 60F10 Large deviations
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