$L_p$-Approximable sequences of vectors and limit distribution of quadratic forms of random variables

Mynbaev, Kairat (2000): $L_p$-Approximable sequences of vectors and limit distribution of quadratic forms of random variables. Published in: Advances in Applied Mathematics , Vol. 26, (2001): pp. 302-329.

Abstract

The properties of $L_2$-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are "two-wing" averages of martingale differences. The results constitute the first significant advancement in the theory of $L_2$-approximable sequences since 1976 when Moussatat introduced a narrower notion of $L_2$-generated sequences. The method relies on a study of certain linear operators in the spaces $L_p$ and $l_p$. A criterion of $L_p$-approximability is given. The results are new even when the weights generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.

• Item Type: MPRA Paper
• Language: English
• Keywords: linear operators in $L_p$ spaces; central limit theorem; quadratic forms of random variables
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

T. W. Anderson, "The Statistical Analysis of Time Series," Wiley, New York, 1971.

T. W. Anderson and D. A. Darling, Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes, {\it Ann. Math. Statist}. 23 (1952), 193-212.

T. W. Anderson and N. Kunitomo, Asymptotic distribution of regression and autoregression coefficients with martingale difference disturbances, {\it J. Multivariate Anal}. 40 (1992), 221-243.

S. Beveridge and C.R. Nelson, A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the "business cycle", {\it J. of Monetary Econom}. 7 (1981), 151-174.

W. Bischoff, A functional central limit theorem for regression models, {\it Ann. Statist}. 26 (1998), 1398-1410.

J. Davidson, "Stochastic Limit Theory: An Introduction for Econometricians," Oxford Univ. Press, Oxford, 1994.

I. C. Gohberg and M. G. Kre{\u\i}n, "Introduction to the Theory of Linear Nonselfadjoint Operators," Amer. Math. Soc., Providence, 1969.

J. D. Hamilton, "Time Series Analysis," Princeton Univ. Press, Princeton, NJ, 1994.

K. Iosida, "Functional Analysis," Springer-Verlag, Berlin, 1965.

D. L. McLeish, Dependent central limit theorems and invariance principles, {\it Ann. Probab}. 2 (1974), 620-628.

H. Milbrodt, Testing stationarity in the mean of autoregressive processes with a nonparametric regression trend, {\it Ann. Statist}. 20 (1992), 1426-1440.

P. W. Millar, Optimal estimation of a general regression function, {\it Ann. Statist}. 10 (1982), 717-740.

M. W. Moussatat, "On the Asymptotic Theory of Statistical Experiments and Some of Its Applications", Ph.D. dissertation, Univ. of California, Berkeley, 1976.

K. T. Mynbaev, Linear models with regressors generated by square-integrable functions, in "7 Escola de S\'eries Temporais e Econometria. Sociedade Brasileira de Econometria" August 6-8, 1997, pp. 80-82.

K. T. Mynbaev, Limits of weighted sums of random variables, Discussion text No.218/2000, Economics Department, Federal University of Cear\'a, Brazil, 2000.

K. T. Mynbaev, Some linear operators in $L_p$ spaces and applications to stochastic limit theory, in "First Latin American Congress of Mathematicians, Instituto de Matem\'atica Pura e Aplicada, Rio de Janeiro, July 31 - August 4, 2000".

K. T. Mynbaev and I. Castelar, The strengths and weaknesses of $L_2$-approximable regressors, Rev. Econ\^ om. Nordeste (in press).

S. Nabeya and K. Tanaka, Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative, {\it Ann. Statist}. 16 (1988), 218-235.

S. Nabeya and K. Tanaka, A general approach to the limiting distribution for estimators in time series regression with nonstable autoregressive errors, {\it Econometrica} 58 (1990), 145-163.

P. C. B. Phillips, Understanding spurious regressions in Econometrics, {\it J. Econometrics} 33 (1986), 311-340.

P. C. B. Phillips, Time series regression with a unit root, {\it Econometrica} 55 (1987), 277-301.

C. Srinivasan and M. Zhou, A central limit theorem for weighted and integrated martingales, {\it Scand. J. Statist}. 22 (1995), 493-504.

K. Tanaka, The Fredholm approach to asymptotic inference on nonstationary and noninvertible time series models, Discussion paper, Hitotsubashi University, Tokyo, 1988.

T. J. Vogelsang, Trend function hypothesis testing in the presence of serial correlation, {\it Econometrica} 66 (1998), 123-148.

Ken-ichi Yoshihara, Central limit theorems for weighted $D[0, 1]$-valued mixing sequences. I. Functional central limit theorems for weighted sums, in "Proceedings, Second World Congress of Nonlinear Analysts, Part 6, Athens, 1966". {\it Nonlinear Analysis}, 30 (1997), 3569-3573.

Ken-ichi Yoshihara, Central limit theorems for weighted $D[0, 1]$-valued mixing sequences. II. Functional central limit theorems for integrated variables, {\it Yokohama Math. J}. 44 (1997), 157-167.