Limiting Sets and Convex Hulls of Samples from Product Measures

Let X₁, X₂, ⋯ be a sequence of independent identically distributed random vectors in Rⁿ (Euclidean n-space). Let the Xᵢ's have a distribution which is a product of n Borel probability measures along an orthogonal set of axes. After sampling m times let Hₘ be the convex hull of ₁, ⋯, Xₘ\. All possible limiting shapes for Hₘ are found along with necessary and sufficient conditions that the limit be obtained.