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CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES

Published online by Cambridge University Press:  03 October 2016

Shin Kanaya
Shin Kanaya*
Affiliation:
Aarhus University, CREATES, and IER
*
*Address correspondence to Shin Kanaya, Department of Economics, Aarhus University, Fuglesangs Alle 4, Aarhus V 8210, Denmark; the Institute of Economic Research (IER), Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8603, Japan; e-mail: skanaya@econ.au.dk
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Abstract

The convergence rates of the sums of α-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 α-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 n; and the other, called the infill assumption, is that it shrinks to zero as n 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.

Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 5 , October 2017 , pp. 1121 - 1153
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

The author thanks Bruce E. Hansen, Bent Nielsen, and Neil Shephard for helpful discussions and suggestions. I would also like to thank the Editor, Peter C. B. Phillips, and three anonymous referees for their constructive and valuable comments, which have greatly improved the original version of this paper. In particular, I would like to express my sincere gratitude to Professor Phillips for generous support and outstanding editorial input into this paper, which were considerable and far in excess of what I could expect. I gratefully acknowledge support from CREATES, Center for Research in Econometric Analysis of Time Series, funded by the Danish National Research Foundation (DNRF78), and from the Danish Council for Independent Research, Social Sciences (grant no. DFF - 4182-00279). Part of this research was conducted while I was visiting the Institute of Economic Research, Kyoto University (under the Joint Research Program of the KIER), the support and hospitality of which are gratefully acknowledged.

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