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Invariance principle in a bilinear model with weak nonlinearity

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Abstract

We consider a series of bilinear sequences

$$X_k^{(n)} = X_{k - 1}^{(n)} + \varepsilon _k + bnX_{k - 1}^{(n)} \varepsilon _{k - 1} , k \geqslant 1$$

, with i.i.d. εk and small bilinearity coefficients bn = βn−1/2 and show that under the standard normalization they converge to a diffusion process Yβ. We provide an explicit form of Yβ, investigate the moments of Yβ, and study the limiting behavior of some other quantities related to X (n)k and important for statistical applications. Bibliography: 5 titles.

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References

  1. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York—London— Sydney (1968).

    Google Scholar 

  2. W. W. Charemza, M. A. Lifshits, and S. B. Makarova, “Conditional testing for unit-root bilinearity in financial time series: some theoretical and empirical results,” J. Econom. Dynam. Control, 29, 1–2 (2004).

    MathSciNet  Google Scholar 

  3. I. I. Gikhman and A. V. Skorokhod, Theory of Random Processes, [in Russian] Vol. 3, Nauka, Moscow (1975).

    Google Scholar 

  4. T. Subba Rao and M. M. Gabr, An Introduction to Bispectral Analysis and Bilinear Time Series Models, Lect. Notes Statist., 24, Springer—Verlag, New York (1984).

    Google Scholar 

  5. G. Terdik, Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis, Lect. Notes Statist., 142, Springer, New York (1999).

    Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    M. A. Lifshits

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  1. M. A. Lifshits
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 97–105.

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Lifshits, M.A. Invariance principle in a bilinear model with weak nonlinearity. J Math Sci 137, 4541–4545 (2006). https://doi.org/10.1007/s10958-006-0247-y

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  • DOI: https://doi.org/10.1007/s10958-006-0247-y

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