Let $Z$ be a random variable with finite moment-generating function $M_Z(\theta):=E[e^{\frac{1}{\theta}Z}]<\infty$ for all $\theta > 0$, and for $\delta \in (0,1]$, define $C_Z^\delta := \inf_{\theta>0}\theta\log M_Z(\theta) - \theta\log(\delta)$, and $SVP_Z^\delta:=E[Z] + \sqrt{(1/\delta)Var[Z]}$. By the Chernoff inequality, one has $P(Z \ge C^\delta_Z) \le \delta$. Also, one notes that if $Z$ is Gaussian, then $C^\delta_Z = SVP_Z^\delta$.

# Question

Are there any interesting bounds on the difference $C^\delta_Z - SVP_Z^\delta$ ? You may assume $Z$ is bounded almost surely, say $P(|Z| \le R)= 1$.