# Convergence of Empirical Processes for Interacting Particle Systems with Applications to Nonlinear Filtering

@article{DelMoral2000ConvergenceOE, title={Convergence of Empirical Processes for Interacting Particle Systems with Applications to Nonlinear Filtering}, author={Pierre Del Moral and Michel Ledoux}, journal={Journal of Theoretical Probability}, year={2000}, volume={13}, pages={225-257} }

In this paper, we investigate the convergence of empirical processes for a class of interacting particle numerical schemes arising in biology, genetic algorithms and advanced signal processing. The Glivenko–Cantelli and Donsker theorems presented in this work extend the corresponding statements in the classical theory and apply to a class of genetic type particle numerical schemes of the nonlinear filtering equation.

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- 2012

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## References

SHOWING 1-10 OF 47 REFERENCES

### Central limit theorem for nonlinear filtering and interacting particle systems

- Mathematics
- 1999

Several random particle systems approaches were recently suggested to solve nonlinear ltering problems numerically. The present analysis is concerned with genetic-type interacting particle systems.…

### Measure-valued processes and interacting particle systems. Application to nonlinear filtering problems

- Mathematics
- 1998

In the paper we study interacting particle approximations of discrete time and measure-valued dynamical systems. These systems have arisen in such diverse scientic disciplines as physics and signal…

### Large deviations for interacting particle systems: Applications to non-linear filtering

- Mathematics, Computer Science
- 1998

### Nonlinear filtering and measure-valued processes

- Business, Mathematics
- 1997

Summary. We construct a sequence of branching particle systems with time and space dependent branching mechanisms whose expectation converges to the solution of the Zakai equation. This gives an…

### Convergence of a Branching Particle Method to the Solution of the Zakai Equation

- MathematicsSIAM J. Appl. Math.
- 1998

A sequence of branching particle systems Un convergent in distribution to the solution of the Zakai equation is constructed, which can be used to solve numerically the filtering problem.

### A particle approximation of the solution of the Kushner–Stratonovitch equation

- Mathematics
- 1998

Abstract. We construct a sequence of branching particle systems αn convergent in measure to the solution of the Kushner–Stratonovitch equation. The algorithm based on this result can be used to solve…

### Large Deviations for Interacting Particle Systems

- Mathematics
- 1999

We consider large systems of interacting particles. We obtain a large deviation principle for the empirical process viewed as a random measure on the path space. The precise rate function is…

### Exact finite-dimensional filters for certain diffusions with nonlinear drift

- Mathematics
- 1981

Let and be independent Wiener processes, and consider the task of estimating a diffusion solving the stochastic DE dx t =f(x t )dt+dw t on the basis of noisy observations defined bydy t =x t dt+db t…