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Kim, Tae-Sung,Ko, Mi-Hwa The Korean Statistical Society 2003 Journal of the Korean Statistical Society Vol.32 No.1
Let{Xt}be an m-dimensional linear process of the form (equation omitted), where{Zt}is a sequence of stationary m-dimensional weakly associated random vectors with EZt = O and E∥Zt∥$^2$<$\infty$. We Prove central limit theorems for multivariate linear processes generated by weakly associated random vectors. Our results also imply a functional central limit theorem.
Ko, Mi-Hwa,Kim, Tae-Sung Korean Mathematical Society 2008 대한수학회논문집 Vol.23 No.1
Let {${\xi}_k,\;k\;{\in}\;{\mathbb{Z}}$} be a strictly stationary associated sequence of H-valued random variables with $E{\xi}_k\;=\;0$ and $E{\parallel}{\xi}_k{\parallel}^2\;<\;{\infty}$ and {$a_k,\;k\;{\in}\;{\mathbb{Z}}$} a sequence of linear operators such that ${\sum}_{j=-{\infty}}^{\infty}\;{\parallel}a_j{\parallel}_{L(H)}\;<\;{\infty}$. For a linear process $X_k\;=\;{\sum}_{j=-{\infty}}^{\infty}\;a_j{\xi}_{k-j}$ we derive that {$X_k} fulfills the functional central limit theorem.
On the Functional Central Limit Theorem of Negatively Associated Processes
Baek Jong Il,Park Sung Tae,Lee Gil Hwan The Korean Statistical Society 2005 Communications for statistical applications and me Vol.12 No.1
A functional central limit theorem is obtained for a stationary linear process of the form $X_{t}= \sum\limits_{j=0}^\infty{a_{j}x_{t-j}}$, where {x_t} is a strictly stationary sequence of negatively associated random variables with suitable conditions and {a_j} is a sequence of real numbers with $\sum\limits_{j=0}^\infty|a_{j}|<\infty$.
Central limit theorem on Chebyshev polynomials
안영호 한국수학교육학회 2014 純粹 및 應用數學 Vol.21 No.4
Let Tl be a transformation on the interval [–1, 1] defined by Chebyshev polynomial of degree l (l≥2), i.e., Tl (cos θ) = cos(lθ). In this paper, we consider Tl as a measure preserving transformation on [–1, 1] with an invariant measure 1/( √(1x²)) dx. We show that if f(x) is a nonconstant step function with finite k-discontinuity points with k < l1, then it satisfies the Central Limit Theorem. We also give an explicit method how to check whether it satisfies the Central LimitTheorem or not in the cases of general step functions with finite discontinuity points.
Kim, Tae-Sung,Ko, Mi-Hwa Korean Mathematical Society 2003 대한수학회보 Vol.40 No.4
A functional central limit theorem is obtained for a stationary linear process of the form $X_{t}=\;{\Sigma_{j=0}}^{\infty}a_{j}{\epsilon}_{t-j}, where {${\in}_{t}$}is a strictly stationary associated sequence of random variables with $E_{{\in}_t}{\;}={\;}0.{\;}E({\in}_t^2){\;}<{\;}{\infty}{\;}and{\;}{a_j}$ is a sequence of real numbers with (equation omitted). A central limit theorem for a stationary linear process generated by stationary associated processes is also discussed.
A functional central limit theorem for positively dependent random vectors
Kim, Tae-Sung,Baek, Jong-Il Korean Mathematical Society 1995 대한수학회논문집 Vol.10 No.3
In this note, we extend the concepts of linearly positive quadrant dependence to the random vectors and prove a functional central limit theorem for positively quadrant dependent sequence of $R^d$-valued or separable Hilbert space valued random elements which satisfy a covariance summability condition. This result is an extension of a functional central limit theorem for weakly associated random vectors of Burton et al. to positive quadrant dependence case.
ON STRICT STATIONARITY OF NONLINEAR ARMA PROCESSES WITH NONLINEAR GARCH INNOVATIONS
Lee, O. The Korean Statistical Society 2007 Journal of the Korean Statistical Society Vol.36 No.2
We consider a nonlinear autoregressive moving average model with nonlinear GARCH errors, and find sufficient conditions for the existence of a strictly stationary solution of three related time series equations. We also consider a geometric ergodicity and functional central limit theorem for a nonlinear autoregressive model with nonlinear ARCH errors. The given model includes broad classes of nonlinear models. New results are obtained, and known results are shown to emerge as special cases.
On the functional central limit theorems for martingale difference random vectors
한광희 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.1
For stationary m-dimensional martingale difference sequences we prove the random functional central limit theorems and propose an almost sure consistent estimator for the limiting covariance matrix.
ON THE FUNCTIONAL CENTRAL LIMIT THEOREMS FOR MARTINGALE DIFFERENCE RANDOM VECTORS
Han, Kwang-Hee Korean Society of Computational and Applied Mathem 2008 Journal of applied mathematics & informatics Vol.26 No.1
For stationary m-dimensional martingale difference sequences we prove the random functional central limit theorems and propose an almost sure consistent estimator for the limiting covariance matrix.
On Strict Stationarity of Nonlinear ARMA Processeswith Nonlinear GARCH Innovations
이외숙 한국통계학회 2007 Journal of the Korean Statistical Society Vol.36 No.3
We consider a nonlinear autoregressive moving average model with non-linear GARCH errors, and nd sucient conditions for the existence of astrictly stationary solution of three related time series equations. We alsoconsider a geometric ergodicity and functional central limit theorem for anonlinear autoregressive model with nonlinear ARCH errors. The givenmodel includes broad classes of nonlinear models. New results are obtained,and known results are shown to emerge as special cases.