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\[
\bar{F}(x)=\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi }}\frac{1}{x}\]
for large \( x \). Finding the inverse of \( \bar{F} \) is difficult, but
fortunately we don't have to as we are only interested in a bound. Note that:
\[
\hat{\bar{F}}(x)\equiv \frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi }}>\bar{F}(x)\]
 which implies that 
\[
\hat{\bar{F}}^{-1}(x)>\bar{F}^{-1}(x)\]
\[
\hat{\bar{F}}^{-1}(\epsilon )=\sqrt{2\ln \frac{1}{\epsilon }-\ln 2\pi }\]


Consequently, 
\[
O\left( \ln \frac{1}{\bar{F}\left( \frac{\bar{F}^{-1}\left( \epsilon \right) }{\gamma }\right) }+\ln \frac{m+1}{\delta }\right) \leq O\left( \ln \frac{1}{\bar{F}\left( \frac{\hat{\bar{F}}^{-1}\left( \epsilon \right) }{\gamma }\right) }+\ln \frac{m+1}{\delta }\right) \]
\[
=O\left( \ln \frac{1}{\bar{F}\left( \frac{\sqrt{2\ln \frac{1}{\epsilon }-\ln 2\pi }}{\gamma }\right) }+\ln \frac{m}{\delta }\right) \]



\[
=O\left( \ln \frac{\sqrt{2\pi }\sqrt{2\ln \frac{1}{\epsilon }-\ln 2\pi }}{\gamma e^{-\frac{2\ln \frac{1}{\epsilon }-\ln 2\pi }{2\gamma ^{2}}}}+\ln \frac{m}{\delta }\right) \]
\[
=O\left( \ln e^{\frac{2\ln \frac{1}{\epsilon }-\ln 2\pi }{2\gamma ^{2}}}+\ln \ln \frac{1}{\epsilon }+\ln \frac{1}{\gamma }+\ln \frac{m}{\delta }\right) \]
\[
=O\left( \frac{\ln \frac{1}{\epsilon }}{\gamma ^{2}}+\ln \ln \frac{1}{\epsilon }+\ln \frac{1}{\gamma }+\ln \frac{m}{\delta }\right) \]
 
\[
=O\left( \frac{\ln \frac{1}{\epsilon }}{\gamma ^{2}}+\ln \frac{1}{\gamma }+\ln \frac{m}{\delta }\right) \]
\[
=O\left( \frac{\ln \frac{1}{\epsilon }}{\gamma ^{2}}+\ln \frac{1}{\gamma }+\ln \frac{m}{\delta }\right) \]


Using the Hoeffding approximation for the tail inequality, we get: 


\[
\Rightarrow e_{Q}\leq \hat{e}_{\gamma }+O\left( \lim _{\epsilon \rightarrow 0}\left( \frac{\frac{1}{\gamma ^{2}}\ln \frac{1}{\epsilon }+\ln \frac{1}{\gamma }+\ln \frac{m}{\delta }}{m}\right) ^{\frac{1}{2}}\right) \]


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