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Weak Convergence of Sample Covariance Matrices to Stochastic Integrals via Martingale Approximations

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Abstract

Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral from zero to one of BdB; where B is vector Brownian motion. For strictly stationary and ergodic sequences, rather than martingale differences, a similar result obtains. In this case, the limit is the same with a constant matrix, of bias terms whose magnitude depends on the serial correlation properties of the sequence. This note gives a simple proof of the result using martingale approximations.

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File URL: http://cowles.econ.yale.edu/P/cd/d08a/d0846.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 846.

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Length: 9 pages
Date of creation: Jul 1987
Date of revision:
Publication status: Published in Econometric Theory (1988), 4: 528-533
Handle: RePEc:cwl:cwldpp:846

Note: CFP 716.
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Keywords: Martingale approximations; stochastic integrals; weak convergence;

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  1. Peter C.B. Phillips, 1985. "Asymptotic Expansions in Nonstationary Vector Autoregressions," Cowles Foundation Discussion Papers 765, Cowles Foundation for Research in Economics, Yale University.
  2. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  3. Park, Joon Y. & Phillips, Peter C.B., 1989. "Statistical Inference in Regressions with Integrated Processes: Part 2," Econometric Theory, Cambridge University Press, vol. 5(01), pages 95-131, April.
  4. Peter C.B. Phillips & Joon Y. Park, 1986. "Statistical Inference in Regressions with Integrated Processes: Part 1," Cowles Foundation Discussion Papers 811R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1987.
  5. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
  6. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, vol. 58(1), pages 113-44, January.
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