[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85477-en":3,"doc-seo-85477-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85477,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Sharp Convergence Bounds for Sums of POD and SPOD Weights","Sharp convergence results for sums of the form Sγ(m)=∑_{v⊆N} γ_v m^{|v|} with product and order dependent (POD) weights establish that the finiteness of Sγ(m) for every m>0 is equivalent to the simple summability condition ∑_{j∈N}Υ_j\u003C∞. For factorial-type POD weights γ_v=(|v|!)^σ∏_{j∈v}j^{−ρ}, the growth of log Sγ(m) is shown to be asymptotically order m^{1/(ρ−σ)} when ρ>σ≥0. The framework extends to smoothness-driven product and order dependent (SPOD) weights.","Sharp convergence bounds for sums of POD and SPOD weights  \nZexin Pan∗  \narXiv :2512 . 13068v2 [math .NA] 13 Jul 2026  \nAbstract  \nThis work analyzes the convergence of sums of the form Sγ (m) = Pv⊆N γvm |v| with product and order dependent (POD) weights γv . We establish that for a nonnegative sequence {Υj | j ∈ N},  \n∞  \nX |v|!m|v| Y Υj \u003C ∞ for all m > 0 if and only if X Υj \u003C ∞ .  \nv⊆N j∈v j=1  \nWe further characterize the growth of Sγ (m) when γv = (|v|!)σ Qj∈v j −ρ and prove that log Sγ (m) is of asymptotic order m 1/(ρ−σ) when ρ > σ ⩾ 0.  \nWe subsequently generalize both the convergence criterion and the asymptotic order of log Sγ (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.  \n1 Introduction  \nThis paper is motivated by tractability analysis in quasi-Monte Carlo (QMC) methods, where one encounters sums of the form  \nSγ (m) =X γvm |v| ,  \nv⊆N  \nwith N = {1, 2 , 3 ,... } and γ = {γv | v ⊆ N} a collection of nonnegative real numbers called weights. Here, the summation over v ⊆ N runs over all finite subsets of N.  \n∗ Institute of Fundamental and Transdisciplinary Research, Zhejiang University, 866 Yuhangtang Road, Xihu District, Hangzhou, Zhejiang Province, 310058, China.([zep002@zju.edu.cn](zep002@zju.edu.cn)).  \nA central problem, arising for instance in the analysis of lattice rules for integration in weighted Korobov spaces [3], is to establish under suitable conditions on γ that  \nSγ (m) \u003C ∞ for all m > 0. (1)  \nProving (1) is key to demonstrating that certain QMC rules achieve error bounds independent of the problem dimension—a property known as strong tractability in information-based complexity [14] .  \nBeyond mere convergence, some applications require precise control over the growth of Sγ (m) as m → ∞ . In the analysis of base-b digital nets [7], for instance, m presents the base-b logarithm of the number of quadrature points N = bm . The goal is often to show Sγ (m) is sub-exponential in m:  \n 1   \nlim sup log Sγ (m) = 0, (2)  \nm→∞ m  \nor equivalently, that for every δ > 0 there exists a constant Cγ,δ \u003C ∞ such that Sγ (m) ⩽ Cγ,δbδm . Establishing (2) for specific weights, such as those in [17, equation (13)], can imply strong tractability of classical sequences like those of Niederreiter in weighted Sobolev spaces.  \nThe convergence properties of Sγ (m) are straightforward for product weights γv = Qj∈v γj . In this case, the sum factors into an infinite product:  \n∞ Sγ (m) =Y (1 + γj m),  \nj=1  \nfrom which it follows that both (1) and (2) are satisfied if P γj \u003C ∞ [9, Lemma 3] .  \nThe analysis becomes more involved for product and order-dependent (POD) weights. These take the general form  \nγv = Γ |v| Y Υj , (3)  \nj∈v  \nparameterized by two nonnegative sequences {Γℓ | ℓ ∈ N∪{0}} and {Υj | j ∈ N} . Here, the factor Γ |v| introduces a dependence on the order (the size |v|) of the subset v. POD weights arise naturally in the analysis of QMC methods for partial differential equations with random coefficients [6, 8, 10, 12, 13] . See [11] for a comprehensive overview.  \nFor the specific yet important class of POD weights where Γ |v| = (|v|!)σ for σ ⩾ 0, a common starting point is the inequality  \n∞ ∞ ℓ  \nX |v|!m|v| Y Υj ⩽ X􀀐mX Υj 􀀑 (4)  \nv⊆N j∈v ℓ=0 j=1  \nestablished in [12, Lemma 6.3] . This bound directly implies the finiteness of Sγ (m) for the case σ = 1, provided mP Υj \u003C 1. Handling the general  \ncase σ  1 then requires additional steps, such as an application of Jensen’s or H¨older’s inequality, as illustrated in the proof of [12, Theorem 6.4] .  \nWhile useful, inequality (4) yields a conservative estimate. It suggests the sum could diverge when m exceeds (P Υj)−1, which severely overestimatesi","cbCaitudmTPXW1TM","https://ap.wps.com/l/cbCaitudmTPXW1TM","pdf",389048,1,14,"English","en",105,"# Introduction\n## Strong tractability and dimension-independent error bounds\n## Product weights and factorization\n## POD weights and factorial growth rates\n## Core bounding theorem (Theorem 1)\n# Proof of Theorem 1\n## Technical lemma (Lemma 1)","[{\"question\":\"What condition on POD weights ensures that Sγ(m) is finite for all m\\u003e0?\",\"answer\":\"Sγ(m) is finite for every m\\u003e0 exactly when the underlying sequence Υ satisfies ∑_{j∈N}Υ_j\\u003c∞.\"},{\"question\":\"How does log Sγ(m) grow for POD weights of the form γ_v=(|v|!)^σ∏_{j∈v}j^{−ρ}?\",\"answer\":\"When ρ\\u003eσ≥0, log Sγ(m) has asymptotic order m^{1/(ρ−σ)}.\"},{\"question\":\"How are the results used in quasi-Monte Carlo (QMC) integration?\",\"answer\":\"The paper applies the theory to show interlaced polynomial lattice rules achieve a dimension-independent convergence rate while removing a commonly imposed assumption in QMC literature.\"}]",1784203891,35,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"sharp-convergence-bounds-for-sums-of-pod-and-spod-weights","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/sharp-convergence-bounds-for-sums-of-pod-and-spod-weights/85477/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What condition on POD weights ensures that Sγ(m) is finite for all m>0?","Question",{"text":75,"@type":76},"Sγ(m) is finite for every m>0 exactly when the underlying sequence Υ satisfies ∑_{j∈N}Υ_j\u003C∞.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does log Sγ(m) grow for POD weights of the form γ_v=(|v|!)^σ∏_{j∈v}j^{−ρ}?",{"text":80,"@type":76},"When ρ>σ≥0, log Sγ(m) has asymptotic order m^{1/(ρ−σ)}.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the results used in quasi-Monte Carlo (QMC) integration?",{"text":84,"@type":76},"The paper applies the theory to show interlaced polynomial lattice rules achieve a dimension-independent convergence rate while removing a commonly imposed assumption in QMC literature.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]