Sharp convergence bounds for sums of POD and SPOD weights
The work provides rigorous theoretical foundations for convergence of weighted sums used in high-dimensional numerical integration, benefiting researchers in quasi-Monte Carlo methods and approximation theory.
This paper provides sharp convergence bounds for sums of POD and SPOD weights, establishing necessary and sufficient conditions for convergence and characterizing asymptotic growth rates. The results are applied to quasi-Monte Carlo integration, showing that interlaced polynomial lattice rules achieve dimension-independent convergence rates without a commonly imposed assumption.
This work analyzes the convergence of sums of the form $S_{\boldsymbolγ}(m)=\sum_{v\subseteq \mathbb{N}}γ_v m^{|v|}$ with product and order dependent (POD) weights $γ_v$. We establish that for a nonnegative sequence $\{Υ_j\mid j\in \mathbb{N}\}$, $$\sum_{v\subseteq \mathbb{N}} |v|! m^{|v|}\prod_{j\in v} Υ_j<\infty \text{ for all } m>0 \text{ if and only if } \sum_{j=1}^\infty Υ_j<\infty.$$ We further characterize the growth of $S_{\boldsymbolγ}(m)$ when $γ_v=(|v|!)^σ\prod_{j\in v}j^{-ρ}$ and prove that $\log S_{\boldsymbolγ}(m)$ is of asymptotic order $m^{1/(ρ-σ)}$ when $ρ>σ\geq 0$. We subsequently generalize both the convergence criterion and the asymptotic order of $\log S_{\boldsymbolγ}(m)$ 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.