An empirical almost sure central limit theorem under the weak dependence assumptions and its application to copula processes
2017 ◽
Vol 71
(1)
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pp. 11
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Let: \(\mathbf{Y=}\left( \mathbf{Y}_{i}\right)\), where \(\mathbf{Y}_{i}=\left( Y_{i,1},...,Y_{i,d}\right)\), \(i=1,2,\dots \), be a \(d\)-dimensional, identically distributed, stationary, centered process with uniform marginals and a joint cdf \(F\), and \(F_{n}\left( \mathbf{x}\right) :=\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}\left(Y_{i,1}\leq x_{1},\dots ,Y_{i,d}\leq x_{d}\right)\) denote the corresponding empirical cdf. In our work, we prove the almost sure central limit theorem for an empirical process \(B_{n}=\sqrt{n}\left( F_{n}-F\right)\) under some weak dependence conditions due to Doukhan and Louhichi. Some application of the established result to copula processes is also presented.
2008 ◽
Vol 12
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pp. 65-87
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2000 ◽
Vol 47
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pp. 61-68
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2012 ◽
Vol 27
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pp. 249-277
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1998 ◽
Vol 326
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pp. 87-92
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2008 ◽
Vol 08
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pp. 173-195
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1999 ◽
Vol 80
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pp. 81-93
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1991 ◽
Vol 38
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pp. 187-203
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