Mynbayev, Kairat and Darkenbayeva, Gulsim (2017): Weak convergence of linear and quadratic forms and related statements on Lpapproximability. Published in: Journal of Mathematical Analysis and Applications , Vol. 473, (2019): pp. 13051319.

PDF
MPRA_paper_101686.pdf Download (168kB)  Preview 
Abstract
In this paper we obtain central limit theorems for quadratic forms of noncausal 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 Lpapproximability of the special sequences which arise in the theory of regressions with slowly varying regressors.
Item Type:  MPRA Paper 

Original Title:  Weak convergence of linear and quadratic forms and related statements on Lpapproximability 
Language:  English 
Keywords:  central limit theorem, Lpapproximability, 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 
Item ID:  101686 
Depositing User:  Kairat Mynbaev 
Date Deposited:  17 Jul 2020 11:50 
Last Modified:  17 Jul 2020 11:50 
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, Lpapproximable 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: Lpapproximability versus Brownian motion, Econometric Theory 25(3) (2009) 748–763. [8] K. Mynbaev, ShortMemory 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. Multivariate 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. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/101686 