In this note we study the error term R_n,L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f:Z⁺→R⁺ satisfying lim_n→∞⁡f(n)=∞ and f(n)=O_ε(e^εn) for every ε>0. Then, the random function t↦[Formula presented]R_n,L(tf(n)) on the interval [0,1] converges in distribution to one- dimensional Brownian motion as n→∞. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers’ mean value formula from [18]. For the individual kth moment of the variable (2f(n))^−1/2R_n,L(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)=O(e^cn) for any fixed c∈(0,c_k), where c_k is a constant depending on k whose optimal value we determine.