Let $Z_1,Z_2,...,Z_n,...$ be a sequence of independent standard normal random variables. Let $X_n=\sum^n_{k=1}\frac{Z_k}{\sqrt{k}}$. Does the limiting distribution of $|X_n|$ exists? If yes, find it; if no, explain.
My try:
Is it my way of solving this question is true? Please correct my mistake.
If the limiting distribution doesn't exists. How to explain such phenomena?
For every $n$, $X_n$ is centered normal with variance $H_n=\sum\limits_{k=1}^n\frac1k$. Since $H_n\to\infty$, $|X_n|$ does not converge in distribution. Since $H_n\sim\log n$, the random variables $$\frac1{\sqrt{\log n}}|X_n|$$ converge in distribution to $|Y|$ where $Y$ is standard normal.