Sankhya: The Indian Journal of Statistics

2003, Volume 65, Pt. 2, 356--388

Central Limit Theorems For Weighted Sums Of A Spatial Process Under A Class Of Stochastic And Fixed Designs

By

S.N. LAHIRI, Iowa State University, Ames, USA

SUMMARY: Let $\{Z(\bfs):\bfs\in\Re^d\}$ be a zero mean stationary random field which is observed at a finite number of locations. % $\bf s_1,\ldots,\bf s_{n}$. In this paper, Central Limit Theorems are proved  for weighted sums of the form %$\sum_{i=1}^{n} $\sum_i \omega_n(\bfs_i)Z(\bfs_i)$ where the locations $\bfs_i$'s are specified by certain stochastic spatial designs driven by sequences of iid random vectors. A complete description of  the effects of the underlying spatial sampling design on the asymptotic variance of the sum is given.  Furthermore, results are also proved for a class of nonrandom spatial designs  based on grids under the {\it mixed increasing-domain} spatial asymptotic structure   that  involves  simultaneous {\it infilling} of increasing domains.

AMS (1991) subject classification. 60F05, 60E20, 60G60.

Key words and phrases. Central limit theorem, infill sampling, increasing-domain asymptotics, long range dependence, random field, strong mixing, stochastic design, spatial design.

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