Sharp convergence bounds for sums of POD and SPOD weights

This work analyzes the convergence of sums of the form $S_{\mathit{\gamma }}(m)={\sum }_{v\subseteq \mathbb{N}}{\gamma }_v{m}^{\mathrm{\mid }v\mathrm{\mid }}$, where ${\gamma }_v$ are product and order dependent (POD) weights. We establish that for nonnegative sequence $\{{\mathrm{{\rm Y}}}_j\mid j\in \mathbb{N}\}$, $$\sum _{v\subseteq \mathbb{N}}\mathrm{\mid }v\mathrm{\mid }!m^{\mathrm{\mid }v\mathrm{\mid }}\prod _{j\in v}{\mathrm{{\rm Y}}}_j<\mathrm{\infty }\text{ for }m>0\text{ if and only if }\sum _{j=1}^{\mathrm{\infty }}{\mathrm{{\rm Y}}}_j<\mathrm{\infty }\mathrm{.}$$ We further characterize the growth of $S_{\mathit{\gamma }}(m)$ when ${\gamma }_v=(\mathrm{\mid }v\mathrm{\mid }!{)}^{\sigma }{\prod }_{j\in v}{j}^{-\rho }$ and prove that $\log S_{\mathit{\gamma }}(m)$ exhibits asymptotic order $m^{1\mathrm{/}(\rho -\sigma )}$ when $\rho >\sigma$. All results are subsequently generalized to smoothness-driven product and order dependent (SPOD) weights.