{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,10,14]],"date-time":"2022-10-14T09:06:48Z","timestamp":1665738408083},"reference-count":50,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2016,10,3]],"date-time":"2016-10-03T00:00:00Z","timestamp":1475452800000},"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":[[2017,10]]},"abstract":"<jats:p>The convergence rates of the sums of <jats:italic>\u03b1<\/jats:italic>-mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and\/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences\/triangular arrays, but their convergence rates were not fully investigated. This study is the first to investigate the convergence rates of the sums of <jats:italic>\u03b1<\/jats:italic>-mixing triangular arrays whose mixing coefficients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is fixed for any sample size <jats:italic>n<\/jats:italic>; and the other, called the <jats:italic>infill<\/jats:italic> assumption, is that it shrinks to zero as <jats:italic>n<\/jats:italic> tends to infinity. Our convergence theorems indicate that an explicit trade-off exists between the rate of convergence and the degree of dependence. While the results under the infill assumption can be seen as a direct extension of those under the fixed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time diffusion process with weak mean reversion, and a near-unit-root process.<\/jats:p>","DOI":"10.1017\/s0266466616000323","type":"journal-article","created":{"date-parts":[[2016,10,3]],"date-time":"2016-10-03T02:23:16Z","timestamp":1475461396000},"page":"1121-1153","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["CONVERGENCE RATES OF SUMS OF <i>\u03b1<\/i>-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES"],"prefix":"10.1017","volume":"33","author":[{"given":"Shin","family":"Kanaya","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,10,3]]},"reference":[{"key":"S0266466616000323_ref3","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1111\/1468-0262.00395","article-title":"Fully nonparametric estimation of scalar diffusion models","volume":"71","author":"Bandi","year":"2003","journal-title":"Econometrica"},{"key":"S0266466616000323_ref50","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1137\/S0040585X97977550","article-title":"On polynomial mixing and convergence rate for stochastic difference and differential equations","volume":"44","author":"Veretennikov","year":"1999","journal-title":"Theory of Probability and its Applications"},{"key":"S0266466616000323_ref23","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1080\/02331888908802205","article-title":"Approximate discrete-time scheme for statistics of diffusion processes","volume":"20","author":"Florens-Zmirou","year":"1989","journal-title":"Statistics"},{"key":"S0266466616000323_ref31","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1017\/S0266466615000079","article-title":"Estimation of stochastic volatility models by nonparametric filtering","volume":"32","author":"Kanaya","year":"2016","journal-title":"Econometric Theory"},{"key":"S0266466616000323_ref30","article-title":"Uniform convergence rates of kernel-based nonparametric estimators for continuous time diffusion processes: A damping function approach","author":"Kanaya","year":"2016","journal-title":"Econometric Theory"},{"key":"S0266466616000323_ref41","unstructured":"Park, J.Y . 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