Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CENTRAL LIMIT TYPE THEOREM FOR WEIGHTED PARTICLE SYSTEMS

  • Cho, Nhan-Sook (Department of General Education Hansung University) ;
  • Kwon, Young-Mee (Department of Computer Science and Information Hansung University)
  • Published : 2004.09.01
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Abstract

We consider a system of particles with locations { $X_{i}$ $^{n}$ (t):t$\geq$0,i＝1,…,n} in $R^{d}$ , time-varying weights { $A_{i}$ $^{n}$ (t) : t $\geq$0,i ＝ 1,…,n} and weighted empirical measure processes $V^{n}$ (t)＝1/n$\Sigma$$_{i=1}$$^{n}$ $A_{i}$ $^{n}$ (t)$\delta$ $X_{i}$ $^{n}$ (t), where $\delta$$_{x}$ is the Dirac measure. It is known that there exists the limit of { $V_{n}$ } in the week* topology on M( $R^{d}$ ) under suitable conditions. If { $X_{i}$ $^{n}$ , $A_{i}$ $^{n}$ , $V^{n}$ } satisfies some diffusion equations, applying Ito formula, we prove a central limit type theorem for the empirical process { $V^{n}$ }, i.e., we consider the convergence of the processes η$_{t}$ $^{n}$ ≡ n( $V^{n}$ -V). Besides, we study a characterization of its limit.t.

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References

  1. T. Chiang, G. Kallianpur and P. Sundar, Propagation of chaos and McKean- Vlasov equation in duals of nuclear spaces, Appl. Math. Optim. 24 (1991), 55–83. https://doi.org/10.1007/BF01447735
  2. S. Ethier and T. Kurtz, Markov processes: Characterization and convergence, Wiley, 1986
  3. C. Graham, Nonlinear Ito-Skorohod equations and martingale problem with discrete jump sets, Stochastic Process. Appl. 40 (1992), 69–82. https://doi.org/10.1016/0304-4149(92)90138-G
  4. A. Joffe and M. Metivier, Weak convergence of sequences of semimartingales with application to multitype branching processes, Adv. in Appl. Probab. 18 (1986), 20–65.
  5. G. Kallianpur and J. Xiong, Asymptotic behavior of a system of interacting nuclear-space-valued stochastic differential equation driven by Poisson random measures, Appl. Math. Optim. 30 (1994), 175–201. https://doi.org/10.1007/BF01189454
  6. T. Kurtz and J. Xiong, Particle representation for a class of nonlinear SPDEs, Stochastic Process. Appl. 83 (1999), 103–126 https://doi.org/10.1016/S0304-4149(99)00024-1
  7. T. Kurtz and J. Xiong, Numerical solutions for a class of SPDEs with applications to filtering, Stochastics in Finite and Infinite Dimension (2000), 233–258
  8. T. Kurtz and J. Xiong, A stochastic evolution equation arising from the fluctuation of a class of interesting particle systems, to be submitted
  9. H. McKean, Propogation of chaos for a class of non-linear parabolic equations, Lecture series in Differential Equations 7 (1967), 41–57
  10. S. Meleard, Asymptotic behavior of some interacting particle systems McKean-Vlasov and Boltzman models, Lecture notes in math. 1627 (1995), 42–95 https://doi.org/10.1007/BFb0093177