[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-608202-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-608202-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","functional-limit-theorems-for-nonstationary-marked-hawkes-processes-in-the-high-intensity-regime","Functional Limit Theorems for Nonstationary Marked Hawkes Processes in the High Intensity Regime","","We study marked Hawkes processes with a non-stationary baseline intensity, a general self-exciting function driven by event ages, and event marks whose distribution varies with the event time. Marks are conditionally independent given event times, but time-dependent in law. An immigration-birth (branching) representation is introduced and an equivalent description via conditional inhomogeneous Poisson processes is derived. In a high intensity regime, functional law of large numbers and functional central limit theorems yield continuous Gaussian process limits characterized by convolutional covariance functionals. Special non-decomposable and multiplicative self-exciting cases are analyzed.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/functional-limit-theorems-for-nonstationary-marked-hawkes-processes-in-the-high-intensity-regime/608202/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/functional-limit-theorems-for-nonstationary-marked-hawkes-processes-in-the-high-intensity-regime/608202.png","ImageObject",300,407,{"name":42,"@type":43},"percent12345","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-01","2026-10-17",true,{"@type":52,"interactionType":53,"userInteractionCount":33},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"How does the model incorporate non-stationarity in marked Hawkes processes?","Question",{"text":62,"@type":63},"Non-stationarity arises from the non-stationary baseline intensity λ0(·), the time-varying distributions of marks Zj(·), and the exciting function H.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"What assumption is made about marks in the studied process?",{"text":67,"@type":63},"Given the sequence of event times, the marks are independent, while each mark’s distribution depends on its associated event time.",{"name":69,"@type":60,"acceptedAnswer":70},"What are the main asymptotic results in the high intensity regime?",{"text":71,"@type":63},"Functional law of large numbers and functional central limit theorems are proved for appropriately scaled Hawkes processes, with limits described by continuous Gaussian processes whose covariance involves convolutional functionals from the branching representation.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},608202,1792460969,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":33,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145},1099529851338,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime  \nBO LI AND GUODONG PANG  \nAbstract. We study marked Hawkes processes with an intensity process which has a non-stationary baseline intensity, a general self-exciting function of event “ages” at each time and marks. The marks are assumed to be conditionally independent given the event times, while the distribution of each mark depends on the event time, that is, time-varying. We 􀀌rst observe an immigration-birth (branching) representation of such a non-stationary marked Hawkes process, and then derive an equivalent representation of the process using the associated conditional inhomogeneous Poisson processes with stochastic intensities. We consider such a Hawkes process in the high intensity regime, where the baseline intensity gets large, while the self-exciting function and distributions of the marks are unscaled, and there is no time-scaling in the scaled Hawkes process. We prove functional law of large numbers and functional central limit theorems (FCLTs) for the scaled Hawkes processes in this asymptotic regime. The limits in the FCLTs are characterized by continuous Gaussian processes with covariance structures expressed with convolution functionals resulting from the branching representation. We also consider the special cases of multiplicative self-exciting functions, and an indicator type of non-decomposable self-exciting functions (including the cases of “ceasing”  \nand “delayed” reproductions as well as their extensions with varying reproduction rates), and study the properties of the limiting Gaussian processes in these special cases.  \n1. Introduction  \nHawkes processes were introduced in [24 , 25 , 27] . They have a self-exciting intensity process that depends on its entire history, and can capture positive auto-correlation, clustering e􀀋ects and overdispersion in the counting/arrival processes. They have been widely used in various applications, for example, 􀀌nance (see the recent reviews in [2, 26]), seismology [39], neuron science [36 , 37 , 9], internet tra􀀎c and queueing [32 , 13 , 21 , 8] . It is usually assumed that the baseline intensity function is constant, and the self-exciting function is a function of the “age” of an event (elapsed time of each event) . In some studies, the self-exciting function can depend on some exogenous randomness, that is, random “marks” associated with each event, see for example [7, 5 , 31 , 28] . This is natural in many applications, since the “marks” carried with each event can a􀀋ect the intensity process, besides the “age” of each event. They are similar to the “noises” in shot noise processes (see, e.g. ,[44, 45, 46, 35] and references therein). The marks are all assumed to be [i.i.d. in](i.i.d. in) the existing literature. However, the marks may be dependent across events as well as dependent on the event times. In this paper, we focus on a particular dependent structure where the marks are conditionally independent given the event times while their distributions depend on the associated event times, that is, a time-varying distribution.  \nSuch non-stationarity may appear in many practical systems. For one example, in [39], a marked Hawkes process is used to model earthquake occurrences and residual analysis, where the marks are magnitude of each occurrence. The distributions of the magnitudes may depend on the epochs of each occurrence. For another example, the e􀀋ect of random forces on a damped harmonic oscillator can be modeled by a marked Hawkes process (or shot noise process [48]), where the forces can have a conditional Gaussian distribution with mean zero and covariance matrix that depends on the time  \nKey words and phrases. Marked Hawkes process, non-stationarity, high intensity asymptotic regime, immigrationbirth representation, functional limit theorems, continuous Gaussian processes.  \n2 BO LI AND GUODONG PANG  \nof random force occurrence; see for example [44, Sec","cbCailmxtoNhsYH1","https://ap.wps.com/l/cbCailmxtoNhsYH1","pdf",577399,49,"English","# Introduction\n## Marked Hawkes processes and non-stationarity sources\n## Model definition and intensity formulation\n## Key limit theorems and limiting Gaussian structure","[{\"question\":\"How does the model incorporate non-stationarity in marked Hawkes processes?\",\"answer\":\"Non-stationarity arises from the non-stationary baseline intensity λ0(·), the time-varying distributions of marks Zj(·), and the exciting function H.\"},{\"question\":\"What assumption is made about marks in the studied process?\",\"answer\":\"Given the sequence of event times, the marks are independent, while each mark’s distribution depends on its associated event time.\"},{\"question\":\"What are the main asymptotic results in the high intensity regime?\",\"answer\":\"Functional law of large numbers and functional central limit theorems are proved for appropriately scaled Hawkes processes, with limits described by continuous Gaussian processes whose covariance involves convolutional functionals from the branching representation.\"}]","Functional Limit Theorems for Nonstationary Marked Hawkes Processes in the High Intensity Regime | PDF",1792223249,123]