• 제목/요약/키워드: martingale differences

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ON THE CONVERGENCE OF SERIES OF MARTINGALE DIFFERENCES WITH MULTIDIMENSIONAL INDICES

  • SON, TA CONG;THANG, DANG HUNG
    • 대한수학회지
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    • 제52권5호
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    • pp.1023-1036
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    • 2015
  • Let {Xn; $n{\succeq}1$} be a field of martingale differences taking values in a p-uniformly smooth Banach space. The paper provides conditions under which the series ${\sum}_{i{\preceq}n}\;Xi$ converges almost surely and the tail series {$Tn={\sum}_{i{\gg}n}\;X_i;n{\succeq}1$} satisfies $sup_{k{\succeq}n}{\parallel}T_k{\parallel}=\mathcal{O}p(b_n)$ and ${\frac{sup_{k{\succeq}n}{\parallel}T_k{\parallel}}{B_n}}{\rightarrow\limits^p}0$ for given fields of positive numbers {bn} and {Bn}. This result generalizes results of A. Rosalsky, J. Rosenblatt [7], [8] and S. H. Sung, A. I. Volodin [11].

An empirical clt for stationary martingale differences

  • Bae, Jong-Sig
    • 대한수학회지
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    • 제32권3호
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    • pp.427-446
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    • 1995
  • Let S be a set and B be a $\sigma$-field on S. We consider $(\Omega = S^Z, T = B^z, P)$ as the basic probability space. We denote by T the left shift on $\Omega$. We assume that P is invariant under T, i.e., $PT^{-1} = P$, and that T is ergodic. We denote by $X = \cdots, X_-1, X_0, X_1, \cdots$ the coordinate maps on $\Omega$. From our assumptions it follows that ${X_i}_{i \in Z}$ is a stationary and ergodic process.

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THE SECOND CENTRAL LIMIT THEOREM FOR MARTINGALE DIFFERENCE ARRAYS

  • Bae, Jongsig;Jun, Doobae;Levental, Shlomo
    • 대한수학회보
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    • 제51권2호
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    • pp.317-328
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    • 2014
  • In Bae et al. [2], we have considered the uniform CLT for the martingale difference arrays under the uniformly integrable entropy. In this paper, we prove the same problem under the bracketing entropy condition. The proofs are based on Freedman inequality combined with a chaining argument that utilizes majorizing measures. The results of present paper generalize those for a sequence of stationary martingale differences. The results also generalize independent problems.

JOINT ASYMPTOTIC DISTRIBUTIONS OF SAMPLE AUTOCORRELATIONS FOR TIME SERIES OF MARTINGALE DIFFERENCES

  • Hwang, S.Y.;Baek, J.S.;Lim, K.E.
    • Journal of the Korean Statistical Society
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    • 제35권4호
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    • pp.453-458
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    • 2006
  • It is well known fact for the iid data that the limiting standard errors of sample autocorrelations are all unity for all time lags and they are asymptotically independent for different lags (Brockwell and Davis, 1991). It is also usual practice in time series modeling that this fact continues to be valid for white noise series which is a sequence of uncorrelated random variables. This paper contradicts this usual practice for white noise. We consider a sequence of martingale differences which belongs to white noise time series and derive exact joint asymptotic distributions of sample autocorrelations. Some implications of the result are illustrated for conditionally heteroscedastic time series.