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.

Note that
$$ \Bbb{E} \exp \left\{ \tfrac{1}{2}\delta X_{i}^{2} \right\} = \frac{1}{\sqrt{1-\delta}}
\quad \text{and} \quad
\Bbb{V} \exp \left\{ \tfrac{1}{2}\delta X_{i}^{2} \right\} = \frac{1}{(1-\delta)^{3/2}} < \infty. $$
Then the right form of the (strong) law of large number would be
$$ \frac{\sqrt{1-\delta}}{n} \sum_{i=1}^{n} e^{\frac{1}{2}\delta X_{i}^{2}} \xrightarrow{n\to\infty} 1 \quad \text{a.s.} $$
