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

Let P be a probability distribution on R (equipped with an Euclidean norm). Let r, s > 0 and assume (αn)n≥1 is an (asymptotically) L(P )-optimal sequence of n-quantizers. In this paper we investigate the asymptotic behavior of the maximal radius sequence induced by the sequence (αn)n≥1 and defined to be for every n ≥ 1, ρ(αn) = max{|a|, a ∈ αn}. We show that if 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 rate of convergence for two classes of distributions with unbounded support : distributions with exponential tails and distributions with polynomial tails.