On the central limit theorem for Toeplitz quadratic forms
Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 653-671
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Let $X(t)$, $t = 0,\pm1,\ldots$, be a real-valued
stationary Gaussian sequence with a spectral density function
$f(\lambda)$. The paper considers the question of
applicability of the central limit theorem (CLT) for a
Toeplitz-type quadratic form $Q_n$ in variables $X(t)$,
generated by an integrable even function $g(\lambda)$.
Assuming that $f(\lambda)$ and $g(\lambda)$ are
regularly varying at $\lambda=0$ of orders $\alpha$ and $\beta$,
respectively, we prove the CLT for the standard normalized
quadratic form $Q_n$ in a critical case
$\alpha+\beta=\frac{1}{2}$.
We also show that the CLT is not valid under
the single condition that the asymptotic variance of $Q_n$
is separated from zero and infinity.
Keywords:
stationary Gaussian sequence, spectral density, Toeplitz-type quadratic forms, central limit theorem, asymptotic variance, slowly varying functions.
@article{TVP_2004_49_4_a1,
author = {A. A. Sahakian and M. S. Ginovyan},
title = {On the central limit theorem for {Toeplitz} quadratic forms},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {653--671},
publisher = {mathdoc},
volume = {49},
number = {4},
year = {2004},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2004_49_4_a1/}
}
A. A. Sahakian; M. S. Ginovyan. On the central limit theorem for Toeplitz quadratic forms. Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 653-671. http://geodesic.mathdoc.fr/item/TVP_2004_49_4_a1/