Asymptotics of all moments of the normalized volume of random convex chains

Establish that for every fixed integer k ≥ 1, as n → ∞, the k-th moment of the normalized area V_n = 2·vol(T_n) of the random convex chain T_n in the triangle T with vertices (0,1), (0,0), and (1,0) satisfies E[V_n^k] = 1 − (2k log n)/(3n) + O(n^{-1}). Here T_n is the convex hull of n i.i.d. uniform points in T together with the two vertices (0,1) and (1,0).

Background

The authors compute exact expressions for E[V_n] and E[V_n^2] and derive their asymptotics as n → ∞, namely E[V_n] = 1 − (2 log n)/(3n) + O(n^{-1}) and E[V_n^2] = 1 − (4 log n)/(3n) + O(n^{-1}). Based on these cases, they formulate a conjecture extending the same pattern to all fixed moments k ≥ 1.

V_n denotes the area of T_n normalized by vol(T), i.e., V_n = 2·vol(T_n). The model T_n is the convex hull of n i.i.d. uniform points in T together with two fixed triangle vertices, forming a random convex chain.