Suppose $X_n$ are independent Gaussian random variables with mean $\mu_n$ and variance $\sigma_n^2$.
Prove that $\{X_n\}$ is stochastically bounded iff $\mu_n$ and $\sigma_n^2$ are bounded sequences.
[If the sequences are bounded then the random variables are stochastically bounded. This is probably the easier side and I could solve it.
May I have some hints for the other direction?]
Best Answer
If $X$ is a standard normal random variable, then $$ \mathbb{P}(X>x)\sim \frac{1}{\sqrt{2\pi}x}e^{-x^2/2}$$ as $x\to\infty$.
Therefore if $X$ is Gaussian with mean $\mu$ and variance $\sigma^2$, then $$ \mathbb{P}(X>x)\sim \frac{\sigma}{\sqrt{2\pi}(x-\mu)}e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$ as $x\to\infty$. I think this should be enough to prove the direction you want.