Rate of convergence of exponential of sample mean to exponential of first moment?

Question

If $X_n \sim N(\mu,\sigma)$ and $T_n = \frac{1}{n}\sum_1^n X_i$

What is the rate of convergence of $e^{T_n}$ to $e^{\mu}$

Answer 1

$T_n \sim N(\mu, \sigma/\sqrt{n})$, and $e^{T_n}$ has a lognormal distribution. Its mean and variance are $$\eqalign{\exp(\mu &+ \sigma^2/(2n))\cr &= \exp(\mu) \left(1 + \dfrac{\sigma^2}{2n} + O\left(\frac{1}{n^2}\right)\right)}$$ and $$\eqalign{\exp(2\mu &+ 2\sigma^2/n) - \exp(2\mu + \sigma^2/n)\cr &= \exp(2\mu) \dfrac{\sigma^2}{n} + O\left(\frac{1}{n^2}\right)}$$ respectively.