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The second result is a central limit theorem for the persistence diagram evaluated on the class of all step functions; this result holds as long as a <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\rho $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c1<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-mixing criterion is satisfied and the distributions of the partial maxima do not decay too slowly. Our results greatly expand those extant in the literature to allow for more fruitful use in statistical applications, beyond idealized settings. Examples of distributions and functions for which the limit theory holds are provided throughout.<\/jats:p>","DOI":"10.1007\/s41468-025-00211-1","type":"journal-article","created":{"date-parts":[[2025,5,25]],"date-time":"2025-05-25T05:09:53Z","timestamp":1748149793000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Convergence of persistence diagrams for discrete time stationary processes"],"prefix":"10.1007","volume":"9","author":[{"given":"Andrew M.","family":"Thomas","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,25]]},"reference":[{"key":"211_CR1","doi-asserted-by":"publisher","first-page":"107509","DOI":"10.1016\/j.patcog.2020.107509","volume":"107","author":"N Atienza","year":"2020","unstructured":"Atienza, N., Gonzalez-D\u00edaz, R., Soriano-Trigueros, M.: On the stability of persistent entropy and new summary functions for topological data analysis. 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