{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T08:50:23Z","timestamp":1782291023927,"version":"3.54.5"},"reference-count":35,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T00:00:00Z","timestamp":1763510400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100478_inline1.png\">\n                          <jats:alt-text content-type=\"machine-generated\">upper L Superscript 1<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                        <\/mml:math>\n                        <jats:tex-math>$L^1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    norm strictly greater than 1 and close to 1 as time goes to infinity. We find that the scaling size determines the scaling behavior of the processes as in Jaisson and Rosenbaum (2015). Specifically, after a suitable rescale of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100478_inline2.png\">\n                          <jats:alt-text content-type=\"machine-generated\">left parenthesis a Subscript upper T Baseline minus 1 right parenthesis divided by upper T e Superscript b Super Subscript upper T Superscript upper T x<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mi>T<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\u2212<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <mml:mrow>\n                            <mml:mo>\/<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mrow>\n                                  <mml:mtext>e<\/mml:mtext>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:msub>\n                                  <mml:mi>b<\/mml:mi>\n                                  <mml:mi>T<\/mml:mi>\n                                <\/mml:msub>\n                                <mml:mi>T<\/mml:mi>\n                                <mml:mi>x<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                        <jats:tex-math>$({a_T-1})\/{T{\\textrm{e}}^{b_TTx}}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , the limit of the sequence of Hawkes processes is deterministic. Also, with another appropriate rescaling of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" content-type=\"simple\" xlink:href=\"S0021900225100478_inline3.png\">\n                          <jats:alt-text content-type=\"machine-generated\">1 divided by upper T squared<\/jats:alt-text>\n                        <\/jats:inline-graphic>\n                        <mml:math xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xmlns:mnf=\"http:\/\/cambridge.org\/core\/manifest\" xmlns:cup=\"http:\/\/contentservices.cambridge.org\" xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/cambridge.org\/core\/metadata\" xmlns:core=\"http:\/\/cambridge.org\/core\" xmlns:c=\"http:\/\/cambridge.org\/core\/content\">\n                          <mml:mn>1<\/mml:mn>\n                          <mml:mrow>\n                            <mml:mo>\/<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:msup>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                        <\/mml:math>\n                        <jats:tex-math>$1\/T^2$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , the sequence converges in law to an integrated Cox\u2013Ingersoll\u2013Ross-like process. This theoretical result may apply to model the recent COVID-19 outbreak in epidemiology and phenomena in social networks.\n                  <\/jats:p>","DOI":"10.1017\/jpr.2025.10047","type":"journal-article","created":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T03:20:01Z","timestamp":1763522401000},"page":"607-622","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Scaling limits for supercritical nearly unstable Hawkes processes"],"prefix":"10.1017","volume":"63","author":[{"given":"Chenguang","family":"Liu","sequence":"first","affiliation":[{"name":"TU Delft"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liping","family":"Xu","sequence":"additional","affiliation":[{"name":"Beihang University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"An","family":"Zhang","sequence":"additional","affiliation":[{"name":"Beihang University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2025,11,19]]},"reference":[{"key":"S0021900225100478_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-49835-5"},{"key":"S0021900225100478_ref29","doi-asserted-by":"publisher","DOI":"10.1145\/3178876.3186108"},{"key":"S0021900225100478_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.arcontrol.2021.02.002"},{"key":"S0021900225100478_ref35","doi-asserted-by":"publisher","DOI":"10.1214\/14-AAP1003"},{"key":"S0021900225100478_ref25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-05265-5"},{"key":"S0021900225100478_ref19","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/58.1.83"},{"key":"S0021900225100478_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2013.04.007"},{"key":"S0021900225100478_ref16","doi-asserted-by":"publisher","DOI":"10.3150\/20-BEJ1235"},{"key":"S0021900225100478_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-68015-4"},{"key":"S0021900225100478_ref32","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2012.12.011"},{"key":"S0021900225100478_ref27","doi-asserted-by":"publisher","DOI":"10.1080\/15326349.2015.1024868"},{"key":"S0021900225100478_ref21","doi-asserted-by":"publisher","DOI":"10.2307\/3212693"},{"key":"S0021900225100478_ref11","doi-asserted-by":"publisher","DOI":"10.1137\/0151029"},{"key":"S0021900225100478_ref7","doi-asserted-by":"publisher","DOI":"10.2307\/1911242"},{"key":"S0021900225100478_ref24","volume-title":"Stochastic Differential Equations and Diffusion Processes","volume":"24","author":"Ikeda","year":"1989"},{"key":"S0021900225100478_ref18","doi-asserted-by":"publisher","DOI":"10.3150\/13-BEJ562"},{"key":"S0021900225100478_ref12","doi-asserted-by":"publisher","DOI":"10.1214\/16-EJS1142"},{"key":"S0021900225100478_ref4","doi-asserted-by":"publisher","DOI":"10.1080\/15326340701645959"},{"key":"S0021900225100478_ref9","volume-title":"An Introduction to the Theory of Point Processes","volume":"I","author":"Daley","year":"2003"},{"key":"S0021900225100478_ref3","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2016.2533397"},{"key":"S0021900225100478_ref5","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1065725193"},{"key":"S0021900225100478_ref13","doi-asserted-by":"publisher","DOI":"10.1214\/14-AAP1089"},{"key":"S0021900225100478_ref33","unstructured":"[33] Zhu, L. 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