Invariance principles for stochastic area and related stochastic integrals |
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Authors: | Svante Janson Michael J Wichura |
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Institution: | Department of Mathematics and Statistics, University of Chicago, Chicago, IL 60637, USA |
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Abstract: | Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Zn, n?1, such that EK2(Z1, Z2)<∞, set An = ∑1?i?j?nK(Zi,Zj), n?1. If the Zn's are two-dimensional and K is the determinant function, An is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the An's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics. |
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Keywords: | 60F05 60H65 60J65 stochastic integral invariance principle stochastic area antisymmetric kernel |
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