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By Theorems 3.3 and 3.4, $$ C_Z^\delta\ge C_{Z;\alpha}^\delta:=\inf_{\theta\in\mathbb R}\Big(\theta+\frac{\|(Z-\theta)_+\|_\alpha}{\delta^{1/\alpha}}\Big) $$ for any $\alpha\in(0,\infty)$, where $\|X\|_\alpha:=(E|X|^\alpha)^{1/\alpha}$. This lower bound on $C_Z^\delta$ may be especially useful when $M_Z(\theta)=Ee^{Z/\theta}=\infty$ for all $\theta>0$, while $EZ_+^\alpha<\infty$ for some $\alpha\in(0,\infty)$, because for all $\alpha\in(0,\infty)$ we still have $$P(Z\ge C_{Z;\alpha}^\delta)\le\delta. $$

By Theorems 3.3 and 3.4, $$ C_Z^\delta\ge C_{Z;\alpha}^\delta:=\inf_{\theta\in\mathbb R}\Big(\theta+\frac{\|(Z-\theta)_+\|_\alpha}{\delta^{1/\alpha}}\Big) $$ for any $\alpha\in(0,\infty)$, where $\|X\|_\alpha:=(E|X|^\alpha)^{1/\alpha}$. This lower bound on $C_Z^\delta$ may be especially useful when $M_Z(\theta)=Ee^{Z/\theta}=\infty$ for all $\theta>0$, while $EZ_+^\alpha<\infty$ for some $\alpha\in(0,\infty)$, because for all $\alpha\in(0,\infty)$ we still have $$P(Z\ge C_{Z;\alpha}^\delta)\le\delta, $$ again by the monotonicity-in-$\alpha$ part of mentioned Theorem 3.4 and formula (3.3) in that paper.

By Theorems 3.3 and 3.4, $$ C_Z^\delta\ge C_{Z;\alpha}^\delta:=\inf_{\theta\in\mathbb R}\Big(\theta+\frac{\|(Z-\theta)_+\|_\alpha}{\delta^{1/\alpha}}\Big) $$ for any $\alpha\in(0,\infty)$, where $\|X\|_\alpha:=(E|X|^\alpha)^{1/\alpha}$. This lower bound on $C_Z^\delta$ may be especially useful when $M_Z(\theta)=\infty$ for all $\theta>0$, while $EZ_+^\alpha<\infty$ for some $\alpha\in(0,\infty)$, because for all $\alpha\in(0,\infty)$ we still have $$P(Z\ge C_{Z;\alpha}^\delta)\le\delta. $$

By Theorems 3.3 and 3.4, $$ C_Z^\delta\ge C_{Z;\alpha}^\delta:=\inf_{\theta\in\mathbb R}\Big(\theta+\frac{\|(Z-\theta)_+\|_\alpha}{\delta^{1/\alpha}}\Big) $$ for any $\alpha\in(0,\infty)$, where $\|X\|_\alpha:=(E|X|^\alpha)^{1/\alpha}$. This lower bound on $C_Z^\delta$ may be especially useful when $M_Z(\theta)=Ee^{Z/\theta}=\infty$ for all $\theta>0$, while $EZ_+^\alpha<\infty$ for some $\alpha\in(0,\infty)$, because for all $\alpha\in(0,\infty)$ we still have $$P(Z\ge C_{Z;\alpha}^\delta)\le\delta. $$

By Theorems 3.3 and 3.4, $$ C_Z^\delta\ge\inf_{\theta\in\mathbb R}\Big(\theta+\frac{\|(Z-\theta)_+\|_\alpha}{\delta^{1/\alpha}}\Big) $$ for any $\alpha\in(0,\infty)$, where $\|X\|_\alpha:=(E|X|^\alpha)^{1/\alpha}$. This lower bound on $C_Z^\delta$ may be especially useful when $M_Z(\theta)=\infty$ for all $\theta>0$.

By Theorems 3.3 and 3.4, $$ C_Z^\delta\ge C_{Z;\alpha}^\delta:=\inf_{\theta\in\mathbb R}\Big(\theta+\frac{\|(Z-\theta)_+\|_\alpha}{\delta^{1/\alpha}}\Big) $$ for any $\alpha\in(0,\infty)$, where $\|X\|_\alpha:=(E|X|^\alpha)^{1/\alpha}$. This lower bound on $C_Z^\delta$ may be especially useful when $M_Z(\theta)=\infty$ for all $\theta>0$, while $EZ_+^\alpha<\infty$ for some $\alpha\in(0,\infty)$, because for all $\alpha\in(0,\infty)$ we still have $$P(Z\ge C_{Z;\alpha}^\delta)\le\delta. $$