Weak convergence of linear and quadratic forms and related statements on Lp-approximability

Mynbayev, Kairat and Darkenbayeva, Gulsim (2017): Weak convergence of linear and quadratic forms and related statements on Lp-approximability. Published in: Journal of Mathematical Analysis and Applications , Vol. 473, (2019): pp. 1305-1319.

Abstract

In this paper we obtain central limit theorems for quadratic forms of non-causal short memory linear processes with independent identically distributed innovations. Nabeya and Tanaka (1988) suggested the format, which links the asymptotic distribution to integral operators. In their approach, integral operators had to have continuous symmetric kernels. Mynbaev (2001) employed the theory of approximations to get rid of the continuity requirement. Here we go one step further by lifting the kernel symmetry condition. Also, we establish Lp-approximability of the special sequences which arise in the theory of regressions with slowly varying regressors.

• Item Type: MPRA Paper
• Language: English
• Keywords: central limit theorem, Lp-approximability, quadratic forms
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C10 - General

References

[1] R. Bhansali, L. Giraitis, P. Kokoszka, Convergence of quadratic forms with nonvanishing diagonal, Statist. Probab. Lett. 77(7) (2007) 726–734.

[2] R. Bhansali, L. Giraitis, P. Kokoszka, Approximations and limit theory for quadratic forms of linear processes, Stochastic Process. Appl. 117(1) (2007) 71–95.

[3] G. Fichtenholz, Differential and Integral Calculus II, Nauka, 1968.

[4] I. Gohberg, M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, 1969.

[5] L. Horvath, Q.-M. Shao, Limit theorems for quadratic forms with applications to Whittle’s estimate, Ann. Appl. Probab. 9(1) (1999) 146–187.

[6] K. Mynbaev, Lp-approximable sequences of vectors and limit distribution of quadratic forms of random variables, Adv. in Appl. Math. 26(4) (2001) 302–329.

[7] K. Mynbaev, Central limit theorems for weighted sums of linear processes: Lp-approximability versus Brownian motion, Econometric Theory 25(3) (2009) 748–763.

[8] K. Mynbaev, Short-Memory Linear Processes and Econometric Applications, Wiley and Sons, 2011. [9] K. Mynbaev, A. Ullah, Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model, J. Multi-variate Anal. 99(2) (2008) 245–277.

[10] S. Nabeya, K. Tanaka, Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative, Ann. Statist. 16(1) (1988) 218–235.

[11] M. Peligrad, S. Utev, Central limit theorem for linear processes, Ann. Probab. 25(2) (1997) 443–456.

[12] M. Peligrad, S. Utev, Central limit theorem for stationary linear processes, Ann. Probab. 34(4) (2006) 1608–1622.

[13] P. Phillips, Regression with slowly varying regressors and nonlinear trends, Econometric Theory 23(4) (2007) 557–614.

[14] E. Seneta, Regularly Varying Functions, Nauka, Moscow, 1985.

[15] K. Tanaka, Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, Wiley and Sons, 1996.

[16] W. Wu, X. Shao, Asymptotic spectral theory for nonlinear time series, Ann. Statist. 35(4) (2007) 1773–1801.