Uniform Lower Bound for Intersection Numbers of $psi$-Classes

We approximate intersection numbers $biglangle psi_1^{d_1}cdots psi_n^{d_n}bigrangle_{g,n}$ on Deligne-Mumfords moduli space $overline{mathcal M}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points by certain closed-form expressions in $d_1,dots,d_n$. Conjecturally, these approximations become asymptotically exact uniformly in $d_i$ when $gtoinfty$ and $n$ remains bounded or grows slowly. In this note we prove a lower bound for the intersection numbers in terms of the above-mentioned approximatingexpressions multiplied by an explicit factor $lambda(g,n)$, which tends to $1$ when $gtoinfty$ and $d_1+dots+d_{n-2}=o(g)$.