Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-rtmr9 Total loading time: 0.198 Render date: 2021-06-13T20:53:42.477Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Poisson superposition processes

Published online by Cambridge University Press:  30 March 2016

Harry Crane
Affiliation:
Rutgers University
Peter Mccullagh
Affiliation:
University of Chicago
Corresponding
E-mail address:
Rights & Permissions[Opens in a new window]

Abstract

Superposition is a mapping on point configurations that sends the n-tuple into the n-point configuration , counted with multiplicity. It is an additive set operation such that the superposition of a k-point configuration in is a kn-point configuration in . A Poisson superposition process is the superposition in of a Poisson process in the space of finite-length -valued sequences. From properties of Poisson processes as well as some algebraic properties of formal power series, we obtain an explicit expression for the Janossy measure of Poisson superposition processes, and we study their law under domain restriction. Examples of well-known Poisson superposition processes include compound Poisson, negative binomial, and permanental (boson) processes.

MSC classification

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2015 

References

Arratia, R., Barbour, A. D. and Tavare, S. (1992). Poisson process approximations for the Ewens sampling formula. Ann. Appl. Prob. 2, 519535.CrossRefGoogle Scholar
Barndorff-Nielsen, O. and Yeo, G. F. (1969). Negative binomial processes. J. Appl. Prob. 6, 633647.CrossRefGoogle Scholar
Cox, D. R. and Isham, V. (1980). Point Processes. Chapman & Hall, London.Google Scholar
Crane, H. (2013a). Permanental partition models and Markovian Gibbs structures. J. Statist. Phys. 153, 698726.CrossRefGoogle Scholar
Crane, H. (2013b). Some algebraic identities for the a-permanent. Linear Algebra Appl. 439, 34453459.CrossRefGoogle Scholar
Donnelly, P. and Grimmett, G. (1993). On the asymptotic distribution of large prime factors. J. London Math. Soc. (2) 47, 395404.CrossRefGoogle Scholar
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87112, 240, 376.CrossRefGoogle ScholarPubMed
Fisher, R. A., Corbet, A. S. and Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. J. Animal Ecol. , 12, 4258.CrossRefGoogle Scholar
Hough, J. B., Krishnapur, M., Peres, Y. and ViráG, B. (2006). Determinantal processes and independence. Prob. Surveys 3, 206229.CrossRefGoogle Scholar
Kingman, J. F. C. (1978). Random partitions in population genetics. Proc. R. Soc. London A 361, 120.CrossRefGoogle Scholar
Kingman, J. F. C. (1978). The representation of partition structures. J. London Math. Soc. (2) 18, 374380.CrossRefGoogle Scholar
Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.Google Scholar
Macchi, O. (1975). The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83122.CrossRefGoogle Scholar
Mccullagh, P. (1987). Tensor Methods in Statistics. Chapman & Hall, London.Google Scholar
Mccullagh, P. and M⊘ller, J. (2006). The permanental process. Adv. Appl. Prob. 38, 873888.CrossRefGoogle Scholar
Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin.Google Scholar
Rubak, E., M⊘ller, J. and Mccullagh, P. (2010). Statistical inference for a class of multivariate negative binomial distributions. Res. Rep. R-2010–10, Department of Mathematical Sciences, Aalborg University.Google Scholar
Streitberg, B. (1990). Lancaster interactions revisited. Ann. Statist. 18, 18781885.CrossRefGoogle Scholar
Valiant, L. G. (1979). The complexity of computing the permanent. Theoret. Comput. Sci. 8, 189201.CrossRefGoogle Scholar
Vere-Jones, D. (1988). A generalization of permanents and determinants. Linear Algebra Appl. 111, 119124.CrossRefGoogle Scholar
Vere-Jones, D. (1997). Alpha-permanents and their applications to mulivariate gamma, negative binomial and ordinary binomial distributions. New Zealand J. Math. 26, 125149.Google Scholar
Wilf, H. S. (2006). Generating functionology, 3rd edn. A K Peters, Wellesey, MA.Google Scholar
You have Access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Poisson superposition processes
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Poisson superposition processes
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Poisson superposition processes
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *