## Quantitative concentration inequalities for empirical measures on non-compact spaces.(English)Zbl 1113.60093

Quantitative concentration estimates for the empirical measure of many independent variables, in transportation distances, are established. Typical estimates are of the form $P\left[\sup_{\| \varphi\| _{\text{Lip}}\leq 1} \left( N^{-1} \sum_{i=1}^N \varphi(X_t^i)- \int \varphi \,d\mu_t \right) > \varepsilon \right] \leq C e^{-\lambda N \varepsilon^2},$ where $$\| \varphi\| _{\text{Lip}}=\sup_{x \neq y} | \varphi(x)-\varphi(y)| d^{-1}(x,y)$$ and $$d(\cdot)$$ stands for the distance in phase space (say Euclidean norm in $$\mathbb{R}^d$$). The core of estimates is based on variants of Sanov’s theorem.
These results are applied to provide some error bounds for particle simulations in a simple mean-field kinetic model for granular media. A system of interacting particles is governed by the system of coupled stochastic differential equations. The tools include coupling arguments, as well as regularity and moment estimates for solutions of differential equations.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60H15 Stochastic partial differential equations (aspects of stochastic analysis)

### Keywords:

transport inequalities; Sanov Theorem
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### References:

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