Fix a lattice $H_{\mathbb Z}$, a polarization $Q$ of fixed weight, and fixed Hodge numbers, and let $D$ be the corresponding period domain of polarized Hodge filtrations on $H_{\mathbb C}=H_{\mathbb Z}\otimes_{\mathbb Z}\mathbb C$ with compact dual $\check D$. Let $d_D$ be the distance function on $D$ induced by the Hodge metric, and let $\Gamma\subset \operatorname{Aut}(H_{\mathbb Z},Q)$ be a monodromy group. Let $\Phi:\Delta^*\to \Gamma\backslash D$ be the period map of an admissible polarized pure variation of Hodge structure over the punctured disc $\Delta^*$ with unipotent local monodromy $T\in\Gamma$. Let $\widetilde\Phi:\mathfrak H\to D$ be a lift along the universal covering map $q:\mathfrak H\to\Delta^*$, $q(z)=e^{2\pi i z}$, normalized by $\widetilde\Phi(z+1)=T\widetilde\Phi(z)$ for every $z\in\mathfrak H$. Let $N=\log T$, so that $T=e^N$ and $N$ is a nilpotent endomorphism of $H_{\mathbb C}$. Then there exist a filtration $F_\infty\in\check D$ and a real number $Y>0$ such that $e^{zN}F_\infty\in D$ for every $z\in\mathfrak H$ with $\operatorname{Im}(z)>Y$, and, on the vertical strip $0\le \operatorname{Re}(z)\le 1$,

\begin{align*} \lim_{\operatorname{Im}(z)\to\infty} d_D\bigl(\widetilde\Phi(z),e^{zN}F_\infty\bigr)=0. \end{align*}

The holomorphic map $z\mapsto e^{zN}F_\infty$ is called the associated nilpotent orbit in the normalization $q(z)=e^{2\pi i z}$.