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Let $\{u(t\,, x)\}_{(t, x)\in \mathbb{R}_+\times \mathbb{R}}$ be the densityof one-dimensional super-Brownian motion starting from Lebesgue measure. Usingthe Laplace functional of super-Brownian motion, we prove that as $N\to\infty$, the normalized spatial integral $N^{-1/2}\int_0^{xN}[u(t\,, z)-1]\rm{d} z$ converges jointly in $(t, x)$ to Brownian sheet in distribution.

Gaussian fluctuation for spatial average of super-Brownian motion