Invariance principles for self-similar set-indexed random fields

Abstract : For a stationary random field $(X_j)_{j\in\Z^d}$ and some measure m on $\R^d$, we consider the set-indexed weighted sum process $S_n(A)=\sum_{j\in\Z^d}m(nA\cap R_j)^\frac12 X_j$, where R_j is the unit cube with lower corner j. We establish a general invariance principle under a p-stability assumption on the X_j's and an entropy condition on the class of sets A. The limit processes are self-similar set-indexed Gaussian processes with continuous sample paths. Using Chentsov's type representations to choose appropriate measures m and particular sets A, we show that these limits can be Lévy (fractional) Brownian fields or (fractional) Brownian sheets.
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Transactions of the American Mathematical Society, American Mathematical Society, 2014, 366 (11), http://www.ams.org/journals/tran/2014-366-11/S0002-9947-2014-06135-7/
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Hermine Biermé, Olivier Durieu. Invariance principles for self-similar set-indexed random fields. Transactions of the American Mathematical Society, American Mathematical Society, 2014, 366 (11), http://www.ams.org/journals/tran/2014-366-11/S0002-9947-2014-06135-7/. <hal-00716437v2>

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