Problem:
I am self-learning about DCT and MCT and related Lemma's. I understand the theorems as constructed, but I am struggling to apply them.
As an example:
$X_n=|Z|^{1/n}$ where $Z\sim N(0,1)$
I am trying to do the following:
a)Identify $X$ such that $X_n\rightarrow X$ a.s
b)Determine if $E(X_n)\rightarrow E(X)$ as $n\rightarrow\infty$
I don't think the answer requires exact calculation of expectation, but seeing that done explicitly might help me get my footing.
Work/attempt: This is not a homework question, so any level of detail is extremely appreciated and welcomed.
This is what I know of MCT/DCT:
Monotone Convergence: $f_1,f_2...f_n\uparrow f$ as $n\rightarrow\infty\Rightarrow \lim_{n\rightarrow\infty}\int f_nd\mu=\int fd\mu$
Dominated Convergence: Given measurable functions $f_1,f_2...$ and $|f_n|\le g$ for some $g$, and if $f_n\rightarrow f \mu-a.e.$, then $\lim_{n\rightarrow\infty}\int|f_n-f|d\mu=0$ and $\lim_{n\rightarrow\infty}\int f_nd\mu=\int fd\mu$
So in my example $X_n$ is a series of nth roots of standard normal RVs. I'm not seeing any monotone convergence, so I look to DCT.
Now the trick is to find a function, $g$, that is always greater than or equal to my series. I'm not sure what $X$ this converges to, which makes it hard to find $g$. My first guess is something like $g=|Z|^n$, but I have no theoretical grounding for this choice, and it's not clear to me that it even meets my criteria.
Help?