A Central Limit Theorem and Large Deviation Principles on ℓnp balls

Let Xn be a random vector uniformly distributed on the unit ball of ℓnp and let En be a random kn-dimensional subspace with knn→λ∈[0,1]. We will study the behavior of the random variable ∥PEnXn∥2 showing that it verifies a Central Limit Theorem as well as some Large Deviation Principles.