## 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. $$P (| Z | le R) = 1$$.