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The permanental process

Published online by Cambridge University Press:  08 September 2016

Peter McCullagh*
Affiliation:
University of Chicago
Jesper Møller*
Affiliation:
Aalborg University
*
Postal address: Department of Statistics, University of Chicago, 5734 University Avenue, Chicago, IL 60637, USA. Email address: [email protected]
∗∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark. Email address: [email protected]
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Abstract

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We extend the boson process first to a large class of Cox processes and second to an even larger class of infinitely divisible point processes. Density and moment results are studied in detail. These results are obtained in closed form as weighted permanents, so the extension is called a permanental process. Temporal extensions and a particularly tractable case of the permanental process are also studied. Extensions of the fermion process along similar lines, leading to so-called determinantal processes, are discussed.

MSC classification

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 2006 

References

Barvinok, A. I. (1996). Two algorithmic results for the traveling salesman problem. Math. Operat. Res. 21, 6584.Google Scholar
Benard, C. and Macchi, O. (1973). Detection and emission processes of quantum particles in a chaotic state. J. Math. Phys. 14, 155167.Google Scholar
Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Prob. 31, 929953.Google Scholar
Clifford, P. and Wei, G. (1993). The equivalence of the Cox process with squared radial Ornstein–Uhlenbeck intensity and the death process in a simple population model. Ann. Appl. Prob. 3, 863873.Google Scholar
Cox, D. R. and Isham, V. (1980). Point Processes. Chapman and Hall, London.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, Elementary Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Diaconis, P. and Evans, S. (2000). Immanants and finite point processes. J. Combinatorial Theory 91, 305321.Google Scholar
Dieudonné, J. (1969). Foundations of Modern Analysis. Academic Press, New York.Google Scholar
Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns, 2nd edn. Arnold, London.Google Scholar
Eisenbaum, N. (2003). On the infinite divisibility of squared Gaussian processes. Prob. Theory Relat. Fields 125, 381392.Google Scholar
Grandell, J. (1976). Doubly Stochastic Poisson Processes (Lecture Notes Math. 529). Springer, Berlin.CrossRefGoogle Scholar
Griffiths, R. C. (1984). Characterization of infinitely divisible multivariate gamma distributions. J. Multivariate Anal. 15, 1320.Google Scholar
Griffiths, R. C. and Milne, R. K. (1987). A class of infinitely divisible multivariate negative binomial distributions. J. Multivariate Anal. 22, 1323.Google Scholar
Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2006). Determinantal processes and independence. Prob. Surveys 3, 206229.Google Scholar
Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12, 134139.Google Scholar
Jerrum, M. R., Sinclair, A. and Vigoda, E. (2004). A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. J. Assoc. Comput. Mach. 51, 671697.Google Scholar
Macchi, O. (1971). Distribution statistique des instants d'émission des photoélectrons d'une lumière thermique. C. R. Acad. Sci. Paris A 272, 437440.Google Scholar
Macchi, O. (1975). The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83122.Google Scholar
Malyshev, V. (1980). Cluster expansions in lattice models of statistical physics and the quantum theory of fields. Russian Math. Surveys 35, 162.Google Scholar
McCullagh, P. (1984). Tensor notation and cumulants of polynomials. Biometrika 71, 461476.Google Scholar
McCullagh, P. and Møller, J. (2005). The permanent process. Tech. Rep. R-2005-29, Department of Mathematical Sciences, Aalborg University.Google Scholar
Minc, H. (1978). Permanents (Encyclopaedia Math. Appl. 6). Addison-Wesley, Reading, MA.Google Scholar
Møller, J. (2003). Shot noise Cox processes. Adv. Appl. Prob. 35, 426.Google Scholar
Møller, J. and Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
Møller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Statist. 25, 451482.Google Scholar
Ripley, B. D. (1977). Modelling spatial patterns (with discussion). J. R. Statist. Soc. B 39, 172212.Google Scholar
Shirai, T. and Takahashi, Y. (2003). Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes. J. Funct. Anal. 205, 414463.Google Scholar
Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55, 923975.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Valiant, L. G. (1979). The complexity of computing the permanent. Theoret. Comput. Sci. 8, 189201.Google Scholar
Van Lieshout, M. N. M. (2000). Markov Point Processes and Their Applications. Imperial College Press, London.Google Scholar
Vere-Jones, D. (1988). A generalization of permanents and determinants. Linear Algebra Appl. 111, 119124.Google Scholar