Improvements of Plachky-Steinebach theorem

We show that the conclusion of Plachky-Steinebach theorem holds true for intervals of the form $\left]\overline{L}_r'(λ),y\right[$, where $\overline{L}_r'(λ)$ is the right derivative (but not necessarily a derivative) of the generalized log-moment generating function $\overline{L}$ at some $λ> 0$ and $y\in\ \left]\overline{L}_r'(λ),+\infty\right]$, under the only two following conditions: $(a)$ $\overline{L}'_r(λ)$ is a limit point of the set $\left\{\overline{L}'_r(t):t>λ\right\}$, $(b)$ $\overline{L}(t_i)$ is a limit for a suitable sequence $(t_i)$. By replacing $\overline{L}_r'(λ)$ by $\overline{L}_r'(λ^+)$, the above result extends verbatim to the case $λ=0$ (replacing $(a)$ by the right continuity of $\overline{L}$ at zero when $\overline{L}_r'(0^+)=-\infty$). No hypothesis is made on $\overline{L}_{]-\infty,λ[}$ (e.g. $\overline{L}_{]-\infty,λ[}$ may be the constant $+\infty$ when $λ=0$); $λ\ge 0$ may be a non-differentiability point of $\overline{L}$ and moreover a limit point of non-differentiability points of $\overline{L}$; $λ=0$ may be a left and right discontinuity point of $\overline{L}$. The map $\overline{L}_{\mid ]λ,λ+\varepsilon[}$ 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. The foregoing is valid for general nets $(μ_α,c_α)$ of Borel probability measures and powers and replacing the intervals $\left]\overline{L}_r'(λ^+),y\right[$ by $]x_α,y_α[$ or $[x_α,y_α]$, where $(x_α,y_α)$ is any net such that $(x_α)$ converges to $\overline{L}_r'(λ^+)$ and $\liminf_αy_α>\overline{L}_r'(λ^+)$.

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