Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d\mid x\geq d)$ is approximately a random variable with generalized Pareto distribution.
Hint: Use the fact that $\operatorname{erf}(x)\approx 1-\frac{1}{\sqrt{x}}e^{-\frac{x^2}{2}}$ for large values of $x$.
My attempt: We recall that the density function for the lognormal distribution is given by $$ f(x)=\frac{1}{x\sigma\sqrt{2\pi}}e^{\frac{-(\log x-\mu)^2}{\sigma}}\:\:\text{ for }x>0. $$ The comulative distribution function for a generalized Pareto random variable is given by $$ G(x)=1-\left(1+\frac{\gamma x}{\theta}\right)^{\frac{-1}{\gamma}}. $$ The objective is to find parameters $\gamma$ and $\theta$ such that $\mathbb{P}(Y\leq y)\approx G(y)$, it is clear that $\gamma$ and $\theta$ will be expressed in terms of $\sigma$, $\mu$ and $d$. My attempt is: \begin{align} \mathbb{P}(Y\leq y) & =1- \frac{1-\int_0^{d+y}\frac{1}{x\sigma\sqrt{2\pi}} e^{\frac{-(\log x-\mu)^2}{\sigma}} \, dy }{1-\int_0^d\frac{1}{x\sigma\sqrt{2\pi}}e^{\frac{-(\log x-\mu)^2}{\sigma}} \, dy} \end{align}
We consider the changge of variable given by $t=\frac{\log x -\mu}{\sqrt{2}\sigma}$, then $dt=\frac{1}{\sqrt{2}\sigma x} \, dx$, so, $dx=\sqrt{2}\sigma x\, dt$. Therefores, we have
\begin{align} \mathbb{P}(Y\leq y) & =1- \frac{1-\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\frac{\log(d+y) -\mu}{\sqrt{2}\sigma}}e^{-t^{2}} \, dy }{1-\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\frac{\log(d) -\mu}{\sqrt{2}\sigma}}e^{-t^{2}} \, dy} \\ &= 1- \frac{1-\frac{1}{\sqrt{\pi}}\int_{-\infty}^0 e^{-t^2} dy- \frac{1}{\sqrt{\pi}}\int_0^{\frac{\log(d+y) -\mu}{\sqrt{2}\sigma}}e^{-t^2} \, dy }{1-\frac{1}{\sqrt{\pi}}\int_{-\infty}^0 e^{-t^2} \, dy-\frac{1}{\sqrt{\pi}}\int_0^{\frac{\log(d) -\mu}{\sqrt{2}\sigma}}e^{-t^2} \, dy} \\ &= 1- \frac{1-\frac{1}{2}- \frac{1}{\sqrt{\pi}}\int_0^{\frac{\log(d+y) -\mu}{\sqrt{2}\sigma}} e^{-t^{2}} \, dy }{1-\frac{1}{2}-\frac{1}{\sqrt{\pi}} \int_0^{\frac{\log(d) -\mu}{\sqrt{2}\sigma}}e^{-t^2} \, dy} \\ &= 1- \frac{\frac{1}{2}- \frac{1}{2} \operatorname{erf} \left(\frac{\log(d+y) -\mu}{\sqrt{2}\sigma}\right) }{\frac{1}{2}- \frac{1}{2} \operatorname{erf}\left(\frac{\log(d) -\mu}{\sqrt{2}\sigma}\right) } \\ &= 1- \frac{1- \operatorname{erf}\left(\frac{\log(d+y) -\mu}{\sqrt{2}\sigma}\right) }{1- \operatorname{erf} \left(\frac{\log(d) -\mu}{\sqrt{2}\sigma}\right) } \\ &\approx 1- \frac{\frac{1}{\sqrt{\frac{\log(d+y) -\mu}{\sqrt{2}\sigma}}}e^{-\frac{(\log(d+y)-\mu)^2}{4\sigma^2}} }{\frac{1}{\sqrt{\frac{\log(d) -\mu}{\sqrt{2}\sigma}}}e^{-\frac{(\log(d)-\mu)^2}{4\sigma^2}} } \leftarrow \text{by hint.}\\ &= 1- \sqrt{\frac{\log(d)-\mu}{\log(d+y)-\mu}}e^{-\frac{ (\log(d+y)-\mu)^2}{4\sigma^2}+\frac{(\log(d)-\mu)^2}{4\sigma^2}}\\ &=1- \sqrt{\frac{\log(d)-\mu}{\log(d+y)-\mu}}e^{\frac{1}{4\sigma^2}(\log(d+y)-\log(d))(\log(y+d)+\log(d)-\mu) }\\ &= 1- \sqrt{\frac{\log(d)-\mu}{\log(d+y)-\mu}}e^{-\frac{(\log(d+y)-\mu)^2}{4\sigma^2}+\frac{(\log(d)-\mu)^2}{4\sigma^2}}\\ &=1- \sqrt{\frac{\log(d)-\mu}{\log(d+y)-\mu}}e^{\frac{1}{4\sigma^2}\log\left(\frac{d+y}{d}\right)\left(\log(dy+d^2)-\mu\right) }\\ \end{align} I do not know how to continue, algebraically I have not been able calibrate the parameters to get what I need.
I ask for your help with this problem, any solution or suggestion will be well received.
