Abstract
Previous results on Edgeworth expansions for sums over a random field are extended to the case where the strong mixing coefficient depends not only on the distance between two sets of random variables, but also on the size of the two sets. The results are applied to the Poisson and the Strauss point processes, giving rise also to local limit results.
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References
Bhattacharya, R. N. and Ranga Rao, R. (1976).Normal Approximation and Asymptotic Expansion, Wiley, New York.
Götze, F. and Hipp, C. (1983). Asymptotic expansions for sums of weakly dependent random vectors,Z. Wahrsch. Verw. Gebiete.,64, 211–240.
Götze, F. and Hipp, C. (1989). Asymptotic expansions for potential functions of i.i.d. random fields,Probab. Theory Related Fields,82, 349–370.
Heinrich, L. (1986). Asymptotic normality of a random point field characteristic,Statistics,17, 453–460.
Heinrich, L. (1987). Asymptotic expansions in the central limit theorem for a special class ofm-dependent random fields I,Math. Nachr.,134, 83–106.
Heinrich, L. (1990). Asymptotic expansions in the central limit theorem for a special class ofm-dependent random fields II—Lattice case,Math. Nachr.,145, 309–327.
Jensen, J. L. (1986). A note on the work of Götze and Hipp concerning asymptotic expansions for sums of weakly dependent random vectors, Memoris, No. 10, Department of Theoretical Statistics, University of Aarhus.
Jensen, J. L. (1988). A note on the work of Götze and Hipp concerning asymptotic expansions for sums of weakly dependent random vectors,Proceedings of the Fourth Prague Symposium on Asymptotic Statistics (eds. P. Mandel and M. Husková), 295–303, Charles University, Prague.
Jensen, J. L. (1989). Asymptotic expansions for strongly mixing Harris recurent Markov chains,Scand. J. Statist.,16, 47–63.
Jensen, J. L. (1993). Asymptotic normality of estimates in spatial point processes,Scand. J. Statist.,20, 97–109.
Petrov, V. V. (1975).Sums of Independent Random Variables, Springer, Berlin.
Strauss, D. J. (1975). A model for clustering,Biometrika,63, 467–475.
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Jensen, J.L. A note on asymptotic expansions for sums over a weakly dependent random field with application to the Poisson and Strauss processes. Ann Inst Stat Math 45, 353–360 (1993). https://doi.org/10.1007/BF00775820
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DOI: https://doi.org/10.1007/BF00775820