Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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:


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 $r<\frac12$


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!

share|cite|improve this question
$K$ represents the Modified Bessel Function of the Second Kind. – Randel Apr 11 at 21:38
up vote 2 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


To answer your third question

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

diverges when $r < \nu/2$

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.