Sharp convergence bounds for sums of POD and SPOD weights

Abstract

This work analyzes the convergence of sums of the form $S_{\mathbf{\gamma }}(m)={\sum }_{v\subseteq \mathbb{N}}{\gamma }_v{m}^{|v|}$ with product and order dependent (POD) weights ${\gamma }_v$. We establish that for a nonnegative sequence $\{{{\rm Y}}_j\mid j\in \mathbb{N}\}$, $$\sum _{v\subseteq \mathbb{N}}|v|!m^{|v|}\prod _{j\in v}{{\rm Y}}_j<\infty \text{ for all }m>0\text{ if and only if }\sum _{j=1}^{\infty }{{\rm Y}}_j<\infty .$$ We further characterize the growth of $S_{\mathbf{\gamma }}(m)$ when ${\gamma }_v=(|v|!{)}^{\sigma }{\prod }_{j\in v}{j}^{-\rho }$ and prove that $\log S_{\mathbf{\gamma }}(m)$ is of asymptotic order $m^{1/(\rho -\sigma )}$ when $\rho >\sigma \ge 0$. We subsequently generalize both the convergence criterion and the asymptotic order of $\log S_{\mathbf{\gamma }}(m)$ to smoothness-driven product and order dependent (SPOD) weights, while noting that a full necessary-and-sufficient analogue remains open. Finally, we apply our theory to quasi-Monte Carlo (QMC) integration, showing that interlaced polynomial lattice rules achieve a dimension-independent convergence rate without a commonly imposed assumption in the QMC literature.