# How to find the characteristic function of a function related with Shannon entropy?

A random variable X is distributed with a known probability distribution $$p(x)$$. Suppose that $$x$$ is sampled in an independent and identically distributed process and with the results $$\vec{x}=(x_1,x_2,...,x_n)$$ of the sample is calculated the function:

$$h(\vec{x})=-\frac{1}{n}\sum_{i=1}^n log (p(x_i))$$

Then the question is what is the characteristic function of $$h$$ ($$\hat{h}$$).

I have face the problem in two ways. The first one, I used the definition and try to calculate the characteristic function as:

$$\hat{h}(\vec{k})=\sum_{\vec{x}} e^{i\vec{k}\vec{x}} h(\vec{x})$$

But I have troubles solving those sums because I don't know the behavior of the probability distribution or each $$x\in X$$.

The second one, I think this problem like the sum of random variables with the form $$y_i=log(p(x_i))$$. However, I don't know if this analysis is factible.

So, I want to know if there is an easy way to solve the problem.

• Isn't it $\sum_{x} e^{ikh(x)} p(x)$ – gunes Mar 8 at 0:16
• Yes, thanks that's solved it – user557651 Mar 8 at 1:35