{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:43:57Z","timestamp":1772297037565,"version":"3.50.1"},"reference-count":32,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"content-version":"unspecified","delay-in-days":4638,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1996,6]]},"abstract":"<jats:p>To obtain consistency results for nonparametric estimators based on stochastic processes relevant in econometrics, we introduce the notions of Hilbert space-valued<jats:italic>L<jats:sub>p<\/jats:sub><\/jats:italic>mixingales and near-epoch dependent arrays, and we prove weak and strong laws of large numbers by using a new exponential inequality for Hilbert (<jats:italic>H<\/jats:italic>) space-valued martingale difference arrays. We follow Andrews (1988,<jats:italic>Econometric Theory<\/jats:italic>4, 458\u2013467), Hansen (1991,<jats:italic>Econometric Theory<\/jats:italic>7, 213\u2013221; 1992,<jats:italic>Econometric Theory<\/jats:italic>8, 421\u2013422), Davidson (1993,<jats:italic>Statistics and Probability Letters<\/jats:italic>16,301\u2013304), and de Jong (1995,<jats:italic>Econometric Theory<\/jats:italic>11, 347\u2013358), extending results for<jats:italic>H<\/jats:italic>= R and improving memory conditions in certain instances. We give as examples consistency results for series and kernel estimators.<\/jats:p>","DOI":"10.1017\/s0266466600006599","type":"journal-article","created":{"date-parts":[[2009,3,10]],"date-time":"2009-03-10T15:41:30Z","timestamp":1236699690000},"page":"284-304","source":"Crossref","is-referenced-by-count":21,"title":["Laws of Large Numbers for Hilbert Space-Valued Mixingales with Applications"],"prefix":"10.1017","volume":"12","author":[{"given":"Xiaohong","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Halbert","family":"White","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2009,2,11]]},"reference":[{"key":"S0266466600006599_ref022","doi-asserted-by":"publisher","DOI":"10.1080\/07474939408800273"},{"key":"S0266466600006599_ref009","doi-asserted-by":"publisher","DOI":"10.1093\/0198774036.001.0001"},{"key":"S0266466600006599_ref021","doi-asserted-by":"publisher","DOI":"10.1007\/BF01212560"},{"key":"S0266466600006599_ref006","unstructured":"Chen X. & White H. (1992c). Asymptotic Properties of Some Projection-Based Robbins-Monro Procedures in a Hilbert Space. Discussion paper 92-46, UCSD Department of Economics."},{"key":"S0266466600006599_ref003","first-page":"2","volume-title":"Lecture Notes in Mathematics","author":"Burkholder","year":"1991"},{"key":"S0266466600006599_ref024","first-page":"829","volume-title":"Annals of Probability","volume":"3","author":"McLeish","year":"1975"},{"key":"S0266466600006599_ref002","doi-asserted-by":"publisher","DOI":"10.2748\/tmj\/1178243286"},{"key":"S0266466600006599_ref018","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600004412"},{"key":"S0266466600006599_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF00542537"},{"key":"S0266466600006599_ref028","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177728190"},{"key":"S0266466600006599_ref029","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177699145"},{"key":"S0266466600006599_ref014","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511526176"},{"key":"S0266466600006599_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600013396"},{"key":"S0266466600006599_ref016","doi-asserted-by":"publisher","DOI":"10.1109\/18.54901"},{"key":"S0266466600006599_ref004","unstructured":"Chen X. & White H. (1992a) Weak and Strong Laws of Large Numbers for Hilbert Space-Valued Mixingales. Discussion paper 92-15, UCSD Department of Economics."},{"key":"S0266466600006599_ref015","volume-title":"A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models","author":"Gallant","year":"1988"},{"key":"S0266466600006599_ref023","doi-asserted-by":"publisher","DOI":"10.2307\/2951508"},{"key":"S0266466600006599_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7152(93)90134-5"},{"key":"S0266466600006599_ref019","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1017\/S0266466600013128","article-title":"Erratum: Strong laws for dependent heterogeneous processes","volume":"8","author":"Hansen","year":"1992","journal-title":"Econometric Theory"},{"key":"S0266466600006599_ref020","doi-asserted-by":"publisher","DOI":"10.2307\/1912775"},{"key":"S0266466600006599_ref005","unstructured":"Chen X. & White H. (1992b) Central Limit and Functional Central Limit Theorems for Hilbert Space-Valued Dependent Processes. Discussion paper 92-35, UCSD Department of Economics."},{"key":"S0266466600006599_ref007","volume-title":"Adaptive Nonparametric Learning with Feedback","author":"Chen","year":"1995"},{"key":"S0266466600006599_ref017","volume-title":"Martingale Limit Theory and Its Applications","author":"Hall","year":"1980"},{"key":"S0266466600006599_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5200-9"},{"key":"S0266466600006599_ref032","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5109\/13049","article-title":"Sequential estimation of a continuous probability density function and mode","volume":"14","author":"Yamato","year":"1971","journal-title":"Bulletin of Mathematical Statistics"},{"key":"S0266466600006599_ref012","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176344949"},{"key":"S0266466600006599_ref025","first-page":"157","article-title":"On regression estimators","volume":"9","author":"Nadaraja","year":"1964","journal-title":"Theory of Probability and Its Applications"},{"key":"S0266466600006599_ref026","first-page":"391","article-title":"Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I","volume":"8","author":"Revesz","year":"1973","journal-title":"Studia Scientiarum Mathematicarum Hungarica"},{"key":"S0266466600006599_ref027","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729586"},{"key":"S0266466600006599_ref030","first-page":"359","article-title":"Smooth regression analysis","volume":"26","author":"Watson","year":"1964","journal-title":"Sankhya"},{"key":"S0266466600006599_ref031","first-page":"258","volume-title":"IEEE Transactions on Information Theory","author":"Wolverton","year":"1969"},{"key":"S0266466600006599_ref011","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600009208"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600006599","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,3]],"date-time":"2021-10-03T00:08:56Z","timestamp":1633219736000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600006599\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,6]]},"references-count":32,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1996,6]]}},"alternative-id":["S0266466600006599"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600006599","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,6]]}}}