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.

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

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