Lower estimates of the Kobayashi distance and limits of complex geodesics

It is proved for a strongly pseudoconvex domain $D$ in ${\mathbb{C}}^d$ with ${\mathcal{C}}^{2,\alpha }$-smooth boundary that any complex geodesic through every two close points of $D$ sufficiently close to $\mathrm{\partial }D$ and whose difference is non-tangential to $\mathrm{\partial }D$ intersect a compact subset of $D$ that depends only on the rate of non-tangentiality. As an application, a lower bound for the Kobayashi distance is obtained.