Characteristic function (Probability)

1. Mar 9, 2005

Zaare

How can I show that if $$\phi(t)$$ is a characteristic function for some distribution, then $$|\phi(t)|^2$$ is also a characteristic function?

2. Mar 9, 2005

Hurkyl

Staff Emeritus
Maybe you could directly work out the distribution for it?

Do you have some sort of theorem on what functions are characteristic functions?

(My text on this stuff is at work. )

3. Mar 10, 2005

Zaare

It is given that $$\phi_{X} (t) = E(e^{itX})$$, and I know that $$E(e^{itX}) = \int_{-\infty}^{\infty}{e^{itx}f_X (x) dx}$$
$$f_X (x)$$ is the probability density function.
I'm trying to find the distribution for it as you said, but I haven't succeeded yet.

Last edited: Mar 10, 2005
4. Mar 10, 2005

matt grime

Right, here are the useful results google gave me in the first hit.

Let Y be another r.v. with distribtuion as -X, then its characteristic function out to be the complex conjugate of phi, I think. At least that seems plausible.

Given two distributions X and Y, the char function of their sum is the product of their char functions.

That is for sure, though I'm not sure about the first bit - you should try the integration

5. Mar 10, 2005

Zaare

I think I can show this, but how does that help me?

X and Y have to be independent for this to be ture, right? And I have to show that this is true for X and X even though X and X are not independent.

Last edited: Mar 10, 2005
6. Mar 10, 2005

matt grime

Yes, so what;s the problem with having Y independent? I didn't say it was -X i said its distribution was -X.

7. Mar 10, 2005

Zaare

Ok, if Y has destribution as -X and X and Y are independent, I can show this:
$$\left| {\phi _X \left( t \right)} \right|^2 = \left| {\phi _X \left( t \right)} \right| \times \left| {\overline {\phi _X \left( t \right)} } \right| = \left| {\phi _X \left( t \right)} \right| \times \left| {\phi _Y \left( t \right)} \right| = \left| {\phi _{X + Y} \left( t \right)} \right|$$
But is it enough? I don't know how to interpret this.

8. Mar 10, 2005

Zaare

By the way, this shows your first statement, right?

$$\left. \begin{array}{l} \phi _Y \left( t \right) = E\left[ {e^{itY} } \right] = E\left[ {e^{ - itX} } \right] = E\left[ {\cos x - i\sin x} \right] = E\left[ {\cos x} \right] - iE\left[ {\sin x} \right] \\ \phi _X \left( t \right) = E\left[ {e^{itX} } \right] = E\left[ {\cos x + i\sin x} \right] = E\left[ {\cos x} \right] + iE\left[ {\sin x} \right] \\ \end{array} \right\} \Rightarrow \underline{\underline {\phi _Y \left( t \right) = \overline {\phi _X \left( t \right)} }}$$

9. Mar 10, 2005

matt grime

Yeah, that's it, though I was thinking of going for a change of variable in the integral, that was all.

If X is an r.v, so is Y, and so is X+Y.... that's sufficient

10. Mar 10, 2005

Zaare

Thank you for all the help. :)