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

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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 $$P(Z\ge C_{Z;\alpha}^\delta)\le\delta, $$ again by the monotonicity-in-$\alpha$ part of mentioned Theorem 3.4 and formula (3.3) in that paper.

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  • $\begingroup$ 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 $P(Z \ge CVaR_Z^{1-\delta}) \le \delta$. $\endgroup$
    – dohmatob
    Commented Apr 15, 2019 at 5:36
  • $\begingroup$ 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! $\endgroup$
    – dohmatob
    Commented Apr 15, 2019 at 5:41
  • $\begingroup$ 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.). $\endgroup$
    – dohmatob
    Commented Apr 15, 2019 at 6:04
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    $\begingroup$ @dohmatob : I am glad that you found this answer to be of use. As for your further question, I think it is a good one; however, addressing it will probably require a full-blown paper (at least), rather than a typical MO answer. $\endgroup$
    – Iosif Pinelis
    Commented Apr 15, 2019 at 14:03
  • $\begingroup$ Ok. Thanks once again. $\endgroup$
    – dohmatob
    Commented Apr 15, 2019 at 15:08

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