$U$-Statistics of Random-Size Samples and Limit Theorems for Systems of Markovian Particles with Non-Poisson Initial Distributions
Feldman, Raisa Epstein ; Rachev, Svetlozar
Ann. Probab., Tome 21 (1993) no. 4, p. 1927-1945 / Harvested from Project Euclid
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Limiting distributions of square-integrable infinite order U-statistics were first studied by Dynkin and Mandelbaum and Mandelbaum and Taqqu. We extend their results to the case of non-Poisson random sample size. Multiple integrals of non-Gaussian generalized fields are constructed to identify the limiting distributions. An invariance principle is also established. We use these results to study the limiting distribution of the amount of charge left in some set by an infinite system of signed Markovian particles when the initial particle density goes to infinity. By selecting the initial particle distribution, we determine the limiting distribution of charge, constructing different non-Gaussian generalized random fields, including Laplace, $\alpha$-stable and their multiple integrals.
Publié le : 1993-10-14
Classification:  Generalized random fields,  infinite particle systems,  Markov processes,  $U$-statistics,  multiple integrals,  Hermite polynomials,  invariance principle,  60G60,  60F17,  60F05,  60E07 
@article{1176989005,
     author = {Feldman, Raisa Epstein and Rachev, Svetlozar},
     title = {$U$-Statistics of Random-Size Samples and Limit Theorems for Systems of Markovian Particles with Non-Poisson Initial Distributions},
     journal = {Ann. Probab.},
     volume = {21},
     number = {4},
     year = {1993},
     pages = { 1927-1945},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176989005}
}

Feldman, Raisa Epstein; Rachev, Svetlozar. $U$-Statistics of Random-Size Samples and Limit Theorems for Systems of Markovian Particles with Non-Poisson Initial Distributions. Ann. Probab., Tome 21 (1993) no. 4, pp.  1927-1945. http://gdmltest.u-ga.fr/item/1176989005/