{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T01:38:10Z","timestamp":1720661890134},"reference-count":33,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2017,12,11]],"date-time":"2017-12-11T00:00:00Z","timestamp":1512950400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["www.cambridge.org"],"crossmark-restriction":true},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2018,12]]},"abstract":"<jats:p>This paper studies the weak convergence of renorming volatilities in a family of GARCH(1,1) models from a functional point of view. After suitable renormalization, it is shown that the limiting distribution is a geometric Brownian motion when the associated top Lyapunov exponent <jats:italic>\u03b3<\/jats:italic> &gt; 0 and is an exponential functional of the maximum process of a Brownian motion when <jats:italic>\u03b3<\/jats:italic> = 0. This indicates that the volatility of the GARCH(1,1)-type model has a completely different random structure according to the sign of <jats:italic>\u03b3<\/jats:italic>. The obtained results further strengthen our understanding of volatilities in GARCH-type models. Simulation studies are carried out to assess our findings.<\/jats:p>","DOI":"10.1017\/s0266466617000470","type":"journal-article","created":{"date-parts":[[2017,12,11]],"date-time":"2017-12-11T09:45:38Z","timestamp":1512985538000},"page":"1370-1382","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["RENORMING VOLATILITIES IN A FAMILY OF GARCH MODELS"],"prefix":"10.1017","volume":"34","author":[{"given":"Dong","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wuqing","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2017,12,11]]},"reference":[{"key":"S0266466617000470_ref6","doi-asserted-by":"crossref","first-page":"956","DOI":"10.1214\/09-EJS452","article-title":"Statistical inference for non-stationary GARCH(p, q) 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