[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83732-en":3,"doc-seo-83732-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83732,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","On Determining the Convergence Rate of an Infinite Product of Stochastic Matrices","By convergent sets of stochastic matrices is meant sets for which every infinite product drawn from any compact subset converges to a rank-one matrix. While exponential convergence is established for compact products inside well-structured families such as scrambling matrices, less is known when the matrices are not all scrambling. This work bounds convergence rates in general convergent sets using submultiplicative seminorms. It shows contractions need not exist in a single seminorm across certain classes, and proves a finite product length forcing contractions for any compact convergent set.","On Determining the Convergence Rate of an Infinite Product of  \nStochastic Matrices*  \nRon Ofir and A. Stephen Morse  \narXiv :2607 .03623v1 [ ee ss . SY] 3 Jul 2026  \nAbstract—By a convergent set is meant a set of stochastic matrices where every infinite product of matrices from every compact subset converges to a rank one matrix. Well-known examples include the set of all scrambling matrices, the set of all stochastic matrices with all diagonal entries positive and a rooted graph, the set of all Sarymsakov matrices, and the set of doubly stochastic matrices with positive diagonal entries and a weakly connected graph. It is known that every infinite product from each compact set of every convergent set converges to its limit exponentially fast, but not much is known about the rate of convergence when not all matrices involved are scrambling matrices. This paper deals with bounding the rate of convergence in convergent sets using submultiplicative seminorms. It is shown that only in some convergent sets all matrices are contractions in the same seminorm, and in particular that this method cannot be used to determine the convergence rate for the class of matrices with positive diagonal entries and a rooted graph. As a second contribution, it is shown that for every compact convergent set and every submultiplicative seminorm, there is a finite number k such that all products of k matrices from the set are contractions in the seminorm. Finally, several open questions are posed for future research.  \nI. INTRODUCTION  \nConsensus protocols are key to many distributed control, estimation, and optimization algorithms [2], [9], [13], [15],[17] . The analysis of such algorithms typically involves the study of the convergence properties of infinite products of stochastic matrices from some subset of the set of all n × n stochastic matrices Sn×n. Two questions invariably arise:  \n1. Does the infinite product of interest converge? 2 . If the infinite product converges, at what rate does convergence take place? The first question is typically addressed by exploiting subsets of Sn×n which are known to be “convergent.” A subset C ⊂ Sn×n is convergent if for each compact subset ¯C ⊂ C , every product of infinitely many matrices from ¯C converges to a rank one matrix, i.e., for every sequence S0 , S1 ,   ∈ ¯C,  \nt St ···S0 = 1c  \nfor some c ∈ R 1 ×n. There are several well-known classes within Sn×n which are convergent including the set of all scrambling matrices 1 S [8], the set R of all stochastic  \n*This work was supported in part by the Air Force Office of Scientific Research, under award numbers FA9550-23-1-0175 and FA9550-25-1-0223 . The work of R. Ofir was partially supported by the Viterbi Fellowship, Technion.  \nBoth authors are with the Department of Electrical and Computer Engineering, Yale University, CT, USA ({ron.ofir,[as.morse](as.morse}@yale.edu)[}](as.morse}@yale.edu)[@yale.edu](as.morse}@yale.edu))  \n1By a scrambling matrix is meant a stochastic matrix with no pair of orthogonal rows.  \nmatrices with positive diagonals and rooted graphs2 [3], the set of all Sarymsakov matrices K [19], and the set D of all doubly stochastic matrices with positive diagonalsand weakly connected graphs [12] . It is known that R is actually the largest convergent subset of the set of all n × n stochastic matrices with positive diagonals [3], and that D is the largest convergent subset of the set of all n × n doubly stochastic matrices with positive diagonals [12] . It is also known that D ⊂ R ⊂ K.  \nOne way to obtain a convergence rate for any infinite product of stochastic matrices from a compact subset ¯C is to use a submultiplicative seminorm defined on a suitably defined subspace of Rn×n containing C 3. For such an approach to work, every matrix in C would have to have a seminorm value less than 1 (i.e., be a contraction in the seminorm) . Such seminorms exist for S and D. For example, consider the well-known coefficient of ergodicity of any mat","cbCais4svqK23YnN","https://ap.wps.com/l/cbCais4svqK23YnN","pdf",275355,3,1,7,"English","en",105,"# Introduction\n## Convergent subsets and convergence questions\n## Convergence-rate bounds via submultiplicative seminorms\n## Limits of a single seminorm and key theorem","[{\"question\":\"What does it mean for a set of stochastic matrices to be convergent?\",\"answer\":\"A subset is convergent if, for every compact subset, every infinite product of matrices from it converges to a rank-one matrix.\"},{\"question\":\"Why are submultiplicative seminorms useful for studying convergence rate?\",\"answer\":\"They can act as “consensus seminorms” that convert bounds on matrix products into contraction behavior, enabling exponential-rate estimates when all matrices contract in the same seminorm.\"},{\"question\":\"What is the main limitation proven for the seminorm approach?\",\"answer\":\"The paper shows there is no single submultiplicative consensus seminorm that makes all matrices in certain convergent classes (specifically related to R ⊂ K) contract with value less than 1.\"}]",1784190062,18,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"on-determining-the-convergence-rate-of-an-infinite-product-of-stochastic-matrices","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/on-determining-the-convergence-rate-of-an-infinite-product-of-stochastic-matrices/83732/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","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 does it mean for a set of stochastic matrices to be convergent?","Question",{"text":75,"@type":76},"A subset is convergent if, for every compact subset, every infinite product of matrices from it converges to a rank-one matrix.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are submultiplicative seminorms useful for studying convergence rate?",{"text":80,"@type":76},"They can act as “consensus seminorms” that convert bounds on matrix products into contraction behavior, enabling exponential-rate estimates when all matrices contract in the same seminorm.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main limitation proven for the seminorm approach?",{"text":84,"@type":76},"The paper shows there is no single submultiplicative consensus seminorm that makes all matrices in certain convergent classes (specifically related to R ⊂ K) contract with value less than 1.","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":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]