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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!

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1 Answer 1

up vote 1 down vote accepted

Let us use this as PDF for the inverse chi-square distribution

$$f(x; \nu) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}$$

The characteristic funcion is

$$\phi_X(t)=\int_0^{+\infty} f(x; \nu) e^{itx}\,\mathrm{d}x$$

$$=\int_0^{+\infty} \frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)} e^{itx}\,\mathrm{d}x$$

Not an easy integral. Probably the function $K$ in your result is the complete elliptic integral of the first kind.

When you calculate the integral using WA you get something with a hypergeometric function, which can probably be converted into a complete elliptic integral of the first kind.

To answer your second question

$$\phi_{\frac{1}{4}(X_1-X_2)}=\frac{1}{4}\phi_{X_1}(t)\phi_{X_2}(-t)$$

To answer your third question

$$E(x^r)=\int_0^{\infty}x^r \cdot f(x; \nu)\mathrm{d}x$$

diverges when $r < \nu/2$

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how do the results follow for sums/differences of inverse chi-squares? –  Justin Oct 25 '12 at 20:55
    
you just have to apply the properties you already mentioned under more thougths. Or is your question how these properties should be proven? –  wnvl Oct 25 '12 at 20:58
    
For your third question it would be good that you added the PDF of your inverse chi-square distribution to the question. Different definitions are possible. –  wnvl Oct 25 '12 at 21:00
    
I guess I am just being dense here. Could you show explicitly what the 2 CFs are (the 1/4 X_1 - X_2) and the sum of X_i's? I don't see the connection between the latter and the normal distribution. I will also add the PDF as you requested. –  Justin Oct 25 '12 at 21:01
    
I updated the answer on the second question for (the 1/4 X_1 - X_2). You just have to replace $\phi$ by the formula with K. For the sum it is similar. –  wnvl Oct 25 '12 at 21:13

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