The Annals of Statistics

Limit theorems for moving averages of discretized processes plus noise

Jean Jacod, Mark Podolskij, and Mathias Vetter

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This paper presents some limit theorems for certain functionals of moving averages of semimartingales plus noise which are observed at high frequency. Our method generalizes the pre-averaging approach (see [Bernoulli 15 (2009) 634–658, Stochastic Process. Appl. 119 (2009) 2249–2276]) and provides consistent estimates for various characteristics of general semimartingales. Furthermore, we prove the associated multidimensional (stable) central limit theorems. As expected, we find central limit theorems with a convergence rate n−1/4, if n is the number of observations.

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Ann. Statist., Volume 38, Number 3 (2010), 1478-1545.

First available in Project Euclid: 24 March 2010

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G44: Martingales with continuous parameter 62M09: Non-Markovian processes: estimation
Secondary: 60G42: Martingales with discrete parameter 62G20: Asymptotic properties

Central limit theorem high-frequency observations microstructure noise quadratic variation semimartingale stable convergence


Jacod, Jean; Podolskij, Mark; Vetter, Mathias. Limit theorems for moving averages of discretized processes plus noise. Ann. Statist. 38 (2010), no. 3, 1478--1545. doi:10.1214/09-AOS756.

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