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REPRESENTATION AND WEAK CONVERGENCE OF STOCHASTIC INTEGRALS WITH FRACTIONAL INTEGRATOR PROCESSES

Published online by Cambridge University Press:  01 December 2009

James Davidson and
Nigar Hashimzade
James Davidson*
Affiliation:
University of Exeter
Nigar Hashimzade
Affiliation:
University of Reading
*
*Address correspondence to James Davidson, Department of Economics, University of Exeter, Exeter EX4 4PU, United Kingdom; e-mail: james.davidson@exeter.ac.uk.
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Abstract

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This paper considers the asymptotic distribution of the sample covariance of a nonstationary fractionally integrated process with the stationary increments of another such process—possibly itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analyzed in a previous paper (Davidson and Hashimzade, 2008), and the construction derived from moving average representations in the time domain. Depending on the values of the long memory parameters and choice of normalization, the limiting integral is shown to be expressible as the sum of a constant and two Itô-type integrals with respect to distinct Brownian motions. In certain cases the latter terms are of small order relative to the former. The mean is shown to match that of the harmonic representation, where the latter is defined, and satisfies the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulas are valid for the full range of the long memory parameters and that they extend to non-Gaussian processes.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 6: Newbold Conference Special Issue , December 2009, pp. 1589 - 1624
Copyright
Copyright © Cambridge University Press 2009

References

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