I am working on a few self-study problems in probability/measure theory and am stuck on characteristic functions. I have the following problem:
Given: $X_1,\ldots,X_n$ are iid inverse chi-square(1) random variables with PDF:
$f(x;\nu)=\frac{2^{-\nu/2}}{\Gamma(\nu/2)}x^{-\nu/2-1}e^{-1/(2x)}$
What is the characteristic function for $\frac14(X_1-X_2)$
What is the characteristic function for $\frac1{n^2}(X_1+\cdots+X_n)$
Is the second example related to the normal distribution?
Lastly, how do I verify that $E(X_1^r)<\infty$ if and only if $r<\frac12$
Ideas/attempts
I found the CF of $X_1$ to be $\frac{2}{\Gamma\frac{\phi}{2}}\left(\frac{-it}{2}\right)^{\frac\phi4}K_{\frac\phi2}\left(\sqrt{-2it}\right)$ just by searching around, but I do not know/understand what $K$ represents.
I am having trouble seeing what the sum or difference of two RVs with the above CF signify, and similarly a sum of them.
For the third part, the "if" is fairly straightforward, but how do I approach showing/proving the "only if"?
More thoughts
I have found the following results for combinations of CFs:
$\phi_{aX+b}(t)=e^{ibt}\phi_X(at),\forall a,b,t\in\mathbb{R}$
If $X_1,...,X_n$ are independent, then $\phi_{X_1+...+X_n}(t)=\prod_{k=1}^n\phi_{X_k}(t)$
If $X_1,X_2$ are independent and have the same distribution, then $\phi_{X_1-X_2}(t)=|\phi_{X_1}(t)|^2$
These facts help get things started, but I'm at a loss of how to continue.
Many thanks!