**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.