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Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with anEuclidean norm $|\cdot|$). Let $ r> 0 $ and let $(\alpha_n)_{n \geq1}$ be an(asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. We investigatethe asymptotic behavior of the maximal radius sequence induced by the sequence$(\alpha_n)_{n \geq1}$ defined for every $n \geq1$ by $\rho(\alpha_n) =\max{|a|, a \in\alpha_n}$. When $\card(\supp(P))$ is infinite, the maximalradius sequence goes to $\sup{|x|, x \in\operatorname{supp}(P)}$ as $n$ goes toinfinity. We then give the exact rate of convergence for two classes ofdistributions with unbounded support: distributions with hyper-exponentialtails and distributions with polynomial tails. In the one-dimensional setting,a sharp rate and constant are provided for distributions with hyper-exponentialtails.

Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers