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Uniform convergence in some limit theorems for multiple particle systems. (English) Zbl 0942.60015

Let \( X_1(t), \dots , X_n(t)\), \(t \geq 0\), be diffusion processes performing a random motion of \(n\) particles starting from a random location \(Y_1,\dots , Y_n.\) Let \(\eta _{n,t}(A) \) be the number of particles in a region \(A \subset R\) at time \(t\) and let \( C_L^{\alpha }(R) \) be the family of Hölder functions \(\varphi : R \to R \) such that for \(x,y \in R\) and \( 0 < \alpha \leq 1 \) it holds \(|\varphi (x) |\leq L, \;|\varphi (y) - \varphi (x) |\leq |y - x |^\alpha .\) A weak convergence of the process \(n^{-1/2}\{\eta _{n,t}(\varphi) - E \eta _{n,t}(\varphi) \} \) to a Gaussian process, uniform both in \(t \in [0, M] \) and \(\varphi \in C^{\alpha }_L (R), \) where \(L, M\) are positive constants, is proved for independent Brownian particles, independent branching Brownian particles and Brownian particles with interactions. Results from asymptotic theory of empirical processes like bracketing central limit theorem are utilized in the solution of the problem.

MSC:

60F05 Central limit and other weak theorems
60J65 Brownian motion
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60F17 Functional limit theorems; invariance principles
62G30 Order statistics; empirical distribution functions
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