Improvements of Plachky–Steinebach theorem

We show that the conclusion of the Plachky–Steinebach theorem holds true for intervals of the form $]\overline{L}_r'(\lambda),y[$, where $\overline{L}_r'(\lambda)$ is the right derivative (but not necessarily a derivative) of the generalized $\mathrm{log}$-moment generating function $\overline{L}$ with some $\lambda> 0$ and $y\in\,]\overline{L}_r'(\lambda),+\infty]$, under only the following two conditions: (a) $\overline{L}'_r(\lambda)$ is a limit point of the set $\{\overline{L}'_r(t)\colon t>\lambda\}$, and (b) $\overline{L}(t_i)$ has limit with $t_i$ belonging to some decreasing sequence converging to $\sup\{t>\lambda\colon\overline{L}_{|]\lambda,t]}\ \text{is affine}\}$. By replacing $\overline{L}_r'(\lambda)$ with $\overline{L}_r'(\lambda^+)$, the above result extends verbatim to the case $\lambda=0$ (replacing (a) by the right continuity of $\overline{L}$ at zero when $\overline{L}_r'(0^+)=-\infty$). No hypothesis is made on $\overline{L}_{]-\infty,\lambda[}$ (for example, $\overline{L}_{]-\infty,\lambda[}$ may be the constant $+\infty$ when $\lambda=0$); $\lambda\ge 0$ may be a nondifferentiability point of $\overline{L}$ and, moreover, a limit point of nondifferentiability points of $\overline{L}$; $\lambda=0$ may be a left and right discontinuity point of $\overline{L}$. The map $\overline{L}_{|]\lambda,\lambda+\varepsilon[}$ may fail to be strictly convex for all $\varepsilon>0$. If we drop the assumption (b), then the same conclusion holds with upper limits in place of limits. Furthermore, the foregoing is valid for general nets $(\mu_\alpha,c_\alpha)$ of Borel probability measures and powers (in place of the sequence $(\mu_n,n^{-1})$) and replacing the intervals $]\overline{L}_r'(\lambda^+),y[$ by $]x_\alpha,y_\alpha[$ or $[x_\alpha,y_\alpha]$, where $(x_\alpha,y_\alpha)$ is any net such that $(x_\alpha)$ converges to $\overline{L}_r'(\lambda^+)$ and $\liminf_\alpha y_\alpha>\overline{L}_r'(\lambda^+)$.