I'm trying to approximate CDF of the integral $$\frac{1}{T}\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$
where $W_t$ is the Wiener process, i.e. $W_t\sim N(0,t)$.
For this I use right Riemann sum: $$\frac{1}{T}\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt\approx\frac{1}{N}\sum_{k=1}^N e^{\sigma W_{\frac{Tk}{N}}+\left(r-\frac{\sigma^2}{2}\right)\frac{Tk}{N}}$$
It is known that $$\sigma W_{\frac{Tk}{N}}+\left(r-\frac{\sigma^2}{2}\right)\frac{Tk}{N}\sim N\left(\left(r-\frac{\sigma^2}{2}\right)\frac{Tk}{N}, \sigma^2\frac{Tk}{N}\right)$$
The $m$-th moment of the random variable $e^{\sigma W_{\frac{Tk}{N}}+\left(r-\frac{\sigma^2}{2}\right)\frac{Tk}{N}}$ is $e^{m\frac{Tk}{N}\left(r-\frac{\sigma^2}{2}\right)+\frac{m^2\sigma^2}{2}\frac{Tk}{N}}$.
The first moment is $e^{\frac{Tk}{N}r}$.
The second moment is $e^{\frac{Tk}{N}\left(2r+\sigma^2\right)}$.
The third moment is $e^{\frac{Tk}{N}\left(3r+3\sigma^2\right)}$.
Now, I'm trying to apply the Berry-Esseen theorem: https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem#Non-identically_distributed_summands (we need to subtract expected values to make expected value equal zero):
$$\left\lvert\frac{\sum_{k=1}^N e^{\sigma W_{\frac{Tk}{N}}+\left(r-\frac{\sigma^2}{2}\right)\frac{Tk}{N}}-\sum_{k=1}^N e^{\frac{Tk}{N}r}}{\sqrt{\sum_{k=1}^N e^{\frac{Tk}{N}\left(2r+\sigma^2\right)}}}-\Phi(x)\right\rvert\le C_0 \frac{\sum_{k=1}^N e^{\frac{Tk}{N}\left(3r+3\sigma^2\right)}}{\left(\sum_{k=1}^N e^{\frac{Tk}{N}\left(2r+\sigma^2\right)}\right)^{\frac{3}{2}}}$$
Each sum can be rewritten as one expression since it is just a geometric sequence: $$SX=\sum_{k=1}^N e^{\frac{Tk}{N}r}=\frac{(e^{rT}-1)e^{\frac{rT}{N}}}{e^{\frac{rT}{N}}-1}$$ $$SX^2=\sum_{k=1}^N e^{\frac{Tk}{N}(2r+\sigma^2)}=\frac{e^{T(2r+\sigma^2)}-1}{e^{\frac{T(2r+\sigma^2)}{N}}-1}+e^{T(2r+\sigma^2)}-1$$ $$SX^3=\sum_{k=1}^N e^{\frac{Tk}{N}(3r+3\sigma^2)}=\frac{e^{T(3r+3\sigma^2)}-1}{e^{\frac{T(3r+3\sigma^2)}{N}}-1}+e^{T(3r+3\sigma^2)}-1$$
Now, the inequality can be rewritten as $$\left\lvert\frac{\sum_{k=1}^N e^{\sigma W_{\frac{Tk}{N}}+\left(r-\frac{\sigma^2}{2}\right)\frac{Tk}{N}}-SX}{\sqrt{SX^2}}-\Phi(x)\right\rvert\le C_0 \frac{SX^3}{\left(SX^2\right)^{\frac{3}{2}}}$$
And finally divide by $N$: $$\left\lvert\frac{\frac{1}{N}\sum_{k=1}^N e^{\sigma W_{\frac{Tk}{N}}+\left(r-\frac{\sigma^2}{2}\right)\frac{Tk}{N}}-\frac{SX}{N}}{\frac{\sqrt{SX^2}}{N}}-\Phi(x)\right\rvert\le C_0 \frac{SX^3}{\left(SX^2\right)^{\frac{3}{2}}}$$
The problem here is that $\frac{SX}{N}$ has finite limit as $N\to\infty$, RHS approaches $0$ as $N\to\infty$, but $\frac{\sqrt{SX^2}}{N}\to 0$ as $N\to\infty$. And LHS is infinite.
Where am I wrong?
Thank you in advance.
