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Comput."],"published-print":{"date-parts":[[2026,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Let\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{I}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    be a fixed admissible ideal on\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{N}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and let\n                    <jats:inline-formula>\n                      <jats:tex-math>$$(\\lambda_n,\\mu_n)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    be a deferred window scheme equipped with positive weights\n                    <jats:inline-formula>\n                      <jats:tex-math>$$w=(w_k)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Using the associated window proportions\n                    <jats:inline-formula>\n                      <jats:tex-math>$$D_n(\\cdot)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and the square\/rectangular pair counts\n                    <jats:inline-formula>\n                      <jats:tex-math>$$Q_n(\\cdot)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$D_{m,n}(\\cdot)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , we define\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{I}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -deferred weighted statistical convergence, its diagonal statistically pre-Cauchy variant, and the corresponding\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{I}}_2$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -deferred weighted frequent Cauchy property on\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{N}}\\times{\\mathbb{N}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . We first prove that\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{I}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -deferred weighted statistical convergence always yields\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{I}}_2$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -deferred weighted frequent Cauchy behavior. Our main upgrade theorem (Theorem 4.4) shows that, for bounded sequences, diagonal pairwise control can be promoted to\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{I}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -deferred weighted statistical convergence provided that the deferred weighted window means form an\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{I}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -convergent sequence; in particular, it suffices that the mean sequence is\n                    <jats:inline-formula>\n                      <jats:tex-math>$${\\mathbb{I}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -Cauchy in\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathbb{R}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Examples show that diagonal testing alone cannot replace the full rectangular scheme and that the mean-coherence assumption is essential.\n                  <\/jats:p>","DOI":"10.1007\/s12190-026-02850-8","type":"journal-article","created":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T10:09:10Z","timestamp":1782727750000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Deferred weighted diagonal upgrade principles for windowed statistical convergence: external ideals and frequent cauchy behavior"],"prefix":"10.1007","volume":"72","author":[{"given":"Mehmet","family":"G\u00fcrdal","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"\u00d6mer","family":"Ki\u015fi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,29]]},"reference":[{"issue":"3","key":"2850_CR1","doi-asserted-by":"publisher","first-page":"413","DOI":"10.2307\/1968524","volume":"33","author":"R.P. 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