# Bounds on difference between "logsumexp" and variance?

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

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.
• Really nice answer; thanks. This answer and the reference paper are particularly interesting because my question is originally motivated by considerations in risk-averse decision-making. Also the classical CVaR also seems to correspond to your $Q_1(Z;1-\delta)$, and therefore FWIW, you have a corollary of the form $P(Z \ge CVaR_Z^{1-\delta}) \le \delta$. Commented Apr 15, 2019 at 5:36
• Question: Do you think one can consider an empirical version of the norms $\|Z\|_\alpha$ in the definition of $C_{Z;\alpha}^\delta$ and still get a similar bound (perhaps with some correction terms) ? I ask this because in the case of SVP, there is such an empirical version called the empirical Bennet inequality (see theorem 4 www0.cs.ucl.ac.uk/staff/M.Pontil/reading/svp-final.pdf). Thanks in advance! Commented Apr 15, 2019 at 5:41
• Put another way, let $Z_1,\ldots,Z_n$ be an i.i.d sample from the distribution of $Z$, and define $\|f(Z)\|_{\alpha,n} := (\frac{1}{n}\sum_{i=1}^n|f(Z_i)|^\alpha)^{1/\alpha}$. Finally, define the random variable $\hat{C}_{n,\alpha}^\delta := \inf_{\theta} \left(\theta + \frac{1}{\delta^{1/\alpha}}\|(Z-\theta)_+ \|_{\alpha,n}\right)$. Is there a reasonable bound of the form (for example!): $P(Z > \hat{C}_{n,\alpha}^\delta - \epsilon(n)) \le \delta$, where $\epsilon_n \rightarrow 0$ very fast (say $\epsilon_n = \mathcal O(1/\sqrt{n})$, etc.). Commented Apr 15, 2019 at 6:04