Sum of exponential of normal random variables Suppose
$X_i \sim N(0,1)$ (independent, identical normal distributions) 
Then by Law of large number,
$$
\sqrt{1-\delta} \frac{1}{n}\sum_i^\infty e^{\frac{\delta}{2}X_i^2} \rightarrow \sqrt{1-\delta} \int e^{\frac{\delta}{2}x^2}\frac{1}{\sqrt{2\pi}}e^{\frac{1}{2}x^2}dx =1 
$$
However, according to simulations, this approximation doesn't seem to work when $\delta$ close to one. Is that ture or just need to run larger samples? Thanks !
Update (6/6): As sos440 mentioned, there's a typo and now fixed.

Shouldn't the variance be
$$
\begin{eqnarray}
\mathbf{V} \bigg( exp\{ \frac{\delta}{2}X_i^2\} \bigg) &=&
\int \bigg( exp\{ \frac{\delta}{2}x^2\} \bigg)^2 \frac{1}{\sqrt{2\pi}} exp \{\frac{-x^2}{2}\} dx\\ && -\bigg(\int exp\{ \frac{\delta}{2}x^2\} \frac{1}{\sqrt{2\pi}} exp \{\frac{-x^2}{2}\}  \bigg)^2
\\
&=& \frac{1}{\sqrt{2\pi}} \int exp \{\frac{-x^2}{2}(1-2\delta)\}dx \\
&& -\bigg( \frac{1}{\sqrt{2\pi}} \int exp\{\frac{-x}{2}(1-\delta)\}dx \bigg)^2\\
&=& \frac{1}{\sqrt{1-2\delta}}-\frac{1}{1-\delta}
\end{eqnarray}
$$
