Central limit theorem for infinitesimal gradient boosting

Let $\hat{F}^{\lambda}_{[t/\lambda]}$ be the stochastic gradient boosting process, let $(\hat{F}_t)_{t\geq 0}$ be its infinitesimal gradient boosting limit, let $\mathcal{T}:L^2_{\mathbf{x}}\to L^2_{\mathbf{x}}$ be the associated operator, and let $\widetilde{T}$ denote the corresponding stochastic operator. Write $\mathbb{D}([0,\infty),L^2_{\mathbf{x}})$ for the Skorokhod path space and $\mathrm{d}_{\hat{F}_t}\mathcal{T}$ for the derivative of $\mathcal{T}$ at $\hat{F}_t$. Central limit theorem. Possibly under stronger assumptions than Assumption A, $\mathcal{T}$ is continuously differentiable as an operator $L^2_{\mathbf{x}}\to L^2_{\mathbf{x}}$, and

$$\frac{1}{\sqrt{\lambda}}\bigl(\hat{F}^{\lambda}_{[t/\lambda]}-\hat{F}_t\bigr)_{t\geq 0}\stackrel{d}{\longrightarrow}\mathscr{F},$$

where convergence holds in distribution on $\mathbb{D}([0,\infty),L^2_{\mathbf{x}})$. The limit $\mathscr{F}=(\mathscr{F}_t)_{t\geq 0}$ is a zero-mean continuous Gaussian process in $L^2_{\mathbf{x}}$ characterized by

$$\mathrm{d}\mathscr{F}_t=\mathrm{d}\mathscr{G}_t+(\mathrm{d}_{\hat{F}_t}\mathcal{T})(\mathscr{F}_t)\,\mathrm{d}t,$$

where $\mathscr{G}$ is a cylindrical Wiener process on $L^2_{\mathbf{x}}$ with covariance

$$\mathbb{E}\left[\langle\mathscr{G}_t,f\rangle\langle\mathscr{G}_s,g\rangle\right]=\int_0^{t\wedge s}\mathbb{E}\left[\langle f,\widetilde{T}(\hat{F}_u)-\mathcal{T}(\hat{F}_u)\rangle\langle g,\widetilde{T}(\hat{F}_u)-\mathcal{T}(\hat{F}_u)\rangle\right].$$

This is a conjectural functional central limit theorem describing the Gaussian fluctuations around the infinitesimal gradient boosting limit; the supplied text gives no resolution, so its status remains open.