Asymptotics of random convex polygonal lines

Abstract

The present paper deals with the limit shape of random plane convex polygonal lines whose edges are independent and identically distributed, with finite first moment. The smoothness of a limit curve depends on some properties of the distribution. The limit curve is determined by the projection of the distribution to the unit circle. This correspondence between limit curves and measures on the unit circle is proved to be a bijection. The emphasis is on limit distributions of deviations of random polygonal lines from a limit curve. Normed differences of Minkowski support functions converge to a Gaussian limit process. The covariance of this process can be found in terms of the initial distribution. In the case of uniform distribution on the unit circle, a formula for the covariance is found. The main result is that a.s. sample functions of the limit process have continuous first derivative satisfying the Holder condition of order a, for any fixed a with 0<a<1/2. Bibliography: 7 titles.