The Annals of Probability

Weak convergence of measure-valued processes and r-point functions

Mark Holmes and Edwin Perkins

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Abstract

We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finite-dimensional distributions. The conditions are convergence of the Fourier transform of the r-point functions and perhaps convergence of the “survival probabilities.” These conditions have recently been shown to hold for a variety of statistical mechanical models, including critical oriented percolation, the critical contact process and lattice trees at criticality, all above their respective critical dimensions.

Article information

Source
Ann. Probab., Volume 35, Number 5 (2007), 1769-1782.

Dates
First available in Project Euclid: 5 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1189000927

Digital Object Identifier
doi:10.1214/009117906000001088

Mathematical Reviews number (MathSciNet)
MR2349574

Zentralblatt MATH identifier
1124.60046

Subjects
Primary: 60G57: Random measures 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F05: Central limit and other weak theorems

Keywords
r-point functions measure-valued processes super-Brownian motion canonical measure critical oriented percolation

Citation

Holmes, Mark; Perkins, Edwin. Weak convergence of measure-valued processes and r -point functions. Ann. Probab. 35 (2007), no. 5, 1769--1782. doi:10.1214/009117906000001088. https://projecteuclid.org/euclid.aop/1189000927


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References

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