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Maxima of moving sums in a Poisson random field

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Hock Peng Chan
Hock Peng Chan*
Affiliation:
National University of Singapore
*
∗ Postal address: Department of Statistics and Applied Probability, Faculty of Science, National University of Singapore, Science Drive 1, 119260, Singapore. Email address: [email protected]
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Abstract

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In this paper we examine the extremal tail probabilities of moving sums in a marked Poisson random field. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. We also provide an alternative representation of the constants of the asymptotic formulae in terms of the occupation measure of the conditional local random field at zero, and extend these representations to the constants of asymptotic tail probabilities of Gaussian random fields.

Keywords

Change of measureGaussian processlarge deviationsmarked Poisson processmoving sumsrandom fieldscan statistics

MSC classification

Primary: 60F10: Large deviations
Secondary: 60G10: Stationary processes 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 41 , Issue 3 , September 2009 , pp. 647 - 663
Copyright
Copyright © Applied Probability Trust 2009 

References

Adler, R. J. (2000). On excursion sets, tube formulas and maxima of random fields. Ann. Appl. Prob. 10, 1–74.Google Scholar
Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Prob. 18, 92–128.Google Scholar
Alm, S. E. (1997). On the distribution of scan statistics of a two-dimensional Poisson process. Adv. Appl. Prob. 29, 1–18.Google Scholar
Anderson, N. H. and Titterington, D. M. (1997). Some methods for investigating spatial clustering with epidemiological applications. J. R. Statist. Soc. A 160, 87–105.Google Scholar
Berman, S. M. (1982). Sojourns and extremes of stationary processes. Ann. Prob. 10, 1–46.Google Scholar
Bickel, P. and Rosenblatt, M. (1973). Two-dimensional random field. In Multivariate Analysis, III, Academic Press, New York, pp. 3–15.Google Scholar
Chan, H. P. and Lai, T. L. (2006). Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices. Ann. Prob. 34, 80–121.Google Scholar
Chan, H. P. and Zhang, N. R. (2007). Scan statistics with weighted observations. J. Amer. Statist. Assoc. 102, 595–602.Google Scholar
Cressie, N. (1993). Statistics for Spatial Data. John Wiley, New York.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics. Springer, New York.Google Scholar
Hogan, M. and Siegmund, D. O. (1986). Large deviations for the maxima of some random fields. Adv. Appl. Math. 7, 2–22.Google Scholar
Kabluchko, Z. and Spodarev, E. (2009). Scan statistics of Lévy noises and marked empirical processes. Adv. Appl. Prob. 41, 13–37.Google Scholar
Loader, C. R. (1991). Large-deviation approximations to the distribution of scan statistics. Adv. Appl. Prob. 23, 751–771.Google Scholar
Naus, J. I. (1965). Clustering of random points in two dimensions. Biometrika 52, 263–267.Google Scholar
Pickands, J. III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–75.Google Scholar
Piterbarg, V. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Transl. Math. Monogr. 148), American Mathematical Society, Providence, RI.Google Scholar
Qualls, C. and Watanabe, H. (1973). Asymptotic properties of Gaussian random fields. Trans. Amer. Math. Soc. 177, 155–171.Google Scholar
Shafie, K., Sigal, B., Siegmund, D. and Worsley, K. J. (2003). Rotation space random fields with an application to fMRI data. Ann. Statist. 31, 1732–1771.CrossRefGoogle Scholar
Siegmund, D. (1985). Sequential Analysis. Springer, New York.Google Scholar
Siegmund, D. O. and Worsley, K. J. (1995). Testing for a signal with unknown location and scale in a stationary Gaussian random field. Ann. Statist. 23, 608–639.Google Scholar
Spivak, M. (1965). Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. W. A. Benjamin, New York.Google Scholar
Storey, J. D. and Siegmund, D. O. (2001). Approximate p-values for local sequence alignments: numerical studies. J. Comput. Biol. 8, 549–556.Google Scholar
Taylor, J. E. (2006). A Gaussian kinematic formula. Ann. Prob. 34, 122–158.Google Scholar
Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Prob. 33, 1362–1396.Google Scholar
Tu, I.-P. (2009). Asymptotic overshoots for arithmetic i.i.d. random variables. Statistica Sinica 19, 315–323.Google Scholar
Tu, I. and Siegmund, D. O. (1999). The maxima of a function of a Markov chain and application to linkage analysis. Adv. Appl. Prob. 31, 510–531.Google Scholar
Woodroofe, M. (1979). Repeated likelihood ratio tests. Biometrika 66, 454–463.Google Scholar