{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T15:12:52Z","timestamp":1773155572838,"version":"3.50.1"},"reference-count":52,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,7,30]],"date-time":"2024-07-30T00:00:00Z","timestamp":1722297600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>In this article, we study the autocorrelation function (ACF), which is a crucial element in time series analysis. We compare the distribution of the ACF, both from a theoretical and empirical point of view. We focus on white noise processes (WN), i.e., uncorrelated, centered, and identically distributed variables, whose ACFs are supposed to be asymptotically independent and converge towards the same normal distribution. But, the study of the sum of the sample ACF contradicts this property. Thus, our findings reveal a deviation of the sample ACF from normality beyond a specific lag. Note that this phenomenon is observed for white noise of varying lengths, and evenforn the residuals of an ARMA(p,q) model. This discovery challenges traditional assumptions of normality in time series modeling. Indeed, when modeling a time series, the crucial step is to validate the estimated model by checking that the associated residuals form white noise. In this study, we show that the widely used portmanteau tests are not completely accurate. Box\u2013Pierce appears to be too conservative, whereas Ljung\u2013Box is too liberal. We suggest an alternative method based on the ACF for establishing the reliability of the portmanteau test and the validity of the estimated model. We illustrate our methodology using money stock data in the USA.<\/jats:p>","DOI":"10.3390\/info15080449","type":"journal-article","created":{"date-parts":[[2024,7,30]],"date-time":"2024-07-30T15:25:23Z","timestamp":1722353123000},"page":"449","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Exploring the Depths of the Autocorrelation Function: Its Departure from Normality"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0897-8663","authenticated-orcid":false,"given":"Hossein","family":"Hassani","sequence":"first","affiliation":[{"name":"The Research Institute of Energy Management and Planning (RIEMP), University of Tehran, Tehran 19395-4697, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2392-1120","authenticated-orcid":false,"given":"Manuela","family":"Royer-Carenzi","sequence":"additional","affiliation":[{"name":"I2M, Aix-Marseille Univ, CNRS, UMR 7373, Centrale Marseille, 13007 Marseille, France"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-1620-5346","authenticated-orcid":false,"given":"Leila Marvian","family":"Mashhad","sequence":"additional","affiliation":[{"name":"Department of Statistics, Payame Noor University, Tehran 19395-4697, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7038-7106","authenticated-orcid":false,"given":"Masoud","family":"Yarmohammadi","sequence":"additional","affiliation":[{"name":"Department of Statistics, Payame Noor University, Tehran 19395-4697, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4109-0690","authenticated-orcid":false,"given":"Mohammad Reza","family":"Yeganegi","sequence":"additional","affiliation":[{"name":"International Institute for Applied Systems Analysis (IIASA), Schlo\u00c3\u0178pl. 1, 2361 Laxenburg, Austria"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Elsaraiti, M., Musbah, H., Merabet, A., and Little, T. 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