Abstract
Let X k ; k ∈ N d be a random field which is asymptotically negative dependent in a certain sense. Define the partial sum process in the usual way so that \(W_n \left( t \right) = \sigma _n^{{\text{ - 1}}} \sum\nolimits_{m \leqq n \cdot t} {\left( {X_m - EX_m } \right)} \quad {\text{for}}\quad t \in \left[ {0,1} \right]^d\), where \(\sigma _n^{\text{2}} = {\text{Var}}\left( {\sum\nolimits_{m \leqq n} {X_m } } \right)\). Under some suitable conditions, we show that W n (·) converges in distribution to a Brownian sheet. Direct consequences of the result are functional central limit theorems for negative dependent random fields. The result is based on some general theorems concerning asymptotically negative dependent random fields, which are of independent interest.
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Zhang, LX. A Functional Central Limit Theorem for Asymptotically Negatively Dependent Random Fields. Acta Mathematica Hungarica 86, 237–259 (2000). https://doi.org/10.1023/A:1006720512467
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DOI: https://doi.org/10.1023/A:1006720512467