Uniform convergence in some limit theorems for multiple particle systems (Q1965885)

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.