pr.probabilidad – Limits on the difference between "logsumexp" and the variance?

Leave $ Z $ Be a random variable with a finite moment generating function. $ M_Z ( theta): = E[e^{frac{1}{theta}Z}]< infty $ for all $ theta> 0 $, and to $ delta in (0,1]$define
$ C_Z ^ delta: = inf _ { theta> 0} theta log M_Z ( theta) – theta log ( delta) $,
Y $ SVP_Z ^ delta: = E[Z] + sqrt {(1 / delta) Var[Z]$. Because of Chernoff's inequality, one has $ P (Z ge C ^ delta_Z) le delta $. Also, it is noted that if $ Z $ It is Gaussian, then $ C ^ delta_Z = SVP_Z ^ delta $.

Are there interesting limits on the difference? $ C ^ delta_Z – SVP_Z ^ delta $ ? You can assume $ Z $ It is delimited almost surely, let's say. $ P (| Z | le R) = 1 $.