Asymptotics of the maximal radius of an Lr-optimal sequence of quantizers

Abstract

Let P be a probability distribution on ℝ^d (equipped with an Euclidean norm |⋅|). Let r > 0 and let (α_n)_n≥1 be an (asymptotically) L^r(P)-optimal sequence of n-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence (α_n)_n≥1 defined for every n ≥ 1 by ρ(α_n) = max{|a|, a ∈ α_n}. When card(supp(P)) is infinite, the maximal radius sequence goes to sup {|x|, x ∈ supp(P)} as n goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails.