ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 41, 52, 60 Zarbaliev, Sakhavat M., Moscow State University, Russia Statistics of convex polygons and approximation of convex curves. Let Π be the set of convex polygons on Z_{+}^{2} starting from the origin and with non-negative tangent of edges, and Π(m_{1}, m_{2}) its subset with the fixed right end (m_{1}, m_{2}). The asymptotics of the ensemble Π(m_{1}, m_{2}) as m_{2}/m_{1} → c has been studied by A. M. Vershik, I. Bárány, Ya. G. Sinai and others. The idea of Sinai’s method is to represent the probability distribution P_{m_{1},m_{2}} on Π(m_{1}, m_{2}) as that induced by a certain distribution Q_{z_{1},z_{2}} on Π (with appropriate parameters z_{1}, z_{2} ≈ 1), conditioned by fixing the right end. We obtain a “perturbation” version of the previous results by letting the parameters z_{1}, z_{2} be functions on Z_{+}^{2}. We also apply Sinai’s method to convex polygon approximations of convex C^{2}-curves. In particular, McMillan’s theorem with correction terms and the CLT are proved.