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

Abstract

Let $P$ be a probability distribution on ${\mathbb{R}}^d$ (equipped with an Euclidean norm $|\cdot |$). Let $r>0$ and let $({\alpha }_n{)}_{n\ge 1}$ 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 $({\alpha }_n{)}_{n\ge 1}$ defined for every $n\ge 1$ by $\rho ({\alpha }_n)=\max |a|,a\in {\alpha }_n$. When $\card(\supp(P))$ is infinite, the maximal radius sequence goes to $\sup |x|,x\in \mathrm{supp}(P)$ 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.

31 pages

References in corpus (2)

• Optimal quantization for the pricing of swing options
• When are Swing options bang-bang and how to use it

Cited by in corpus (1)

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