Limiting Sets and Convex Hulls of Samples from Product Measures

Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent identically distributed random vectors in $R^n$ (Euclidean $n$-space). Let the $X_i$'s have a distribution which is a product of $n$ Borel probability measures along an orthogonal set of axes. After sampling $m$ times let $H_m$ be the convex hull of $\{X_1, \cdots, X_m\}$. All possible limiting shapes for $H_m$ are found along with necessary and sufficient conditions that the limit be obtained.