Convergence rates of sums of α-mixing triangular arrays: with an application to non-parametric drift function estimation of continuous-time processes

Shin Kanaya

No 947, KIER Working Papers from Kyoto University, Institute of Economic Research

Abstract: The convergence rates of the sums of α-mixing (or strongly mixing) triangular arrays of het- erogeneous 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 non- existence 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.

Keywords: Law of large numbers; rate of convergence; α-mixing triangular array; infill asymp- totics; kernel estimation. (search for similar items in EconPapers)
JEL-codes: C14 C22 C58 (search for similar items in EconPapers)
Pages: 30pages
Date: 2016-08
New Economics Papers: this item is included in nep-ecm and nep-ets
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Related works:
Journal Article: CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES (2017)
Working Paper: Convergence rates of sums of α-mixing triangular arrays: with an application to non-parametric drift function estimation of continuous-time processes (2016)
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