# Questions tagged [characteristic-functions]

Questions about characteristic functions, of a set (which gives $1$ if the element is on the set and $0$ otherwise) or of a random variable (its Fourier transform). Do not use this tag if you are asking about the method of characteristics in PDE or the characteristic polynomial in linear algebra.

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### Chf. of a sum of independent r.v. is the product of chfs. Is the converse true?

Let suppose that we have a random variables $X$ and $Y$ whose chfs. can be written as: $$\forall u \in\mathbb{R},\quad\Phi_X(u) = f(u)\Phi_Y(u).$$ Does it mean that we can write $X=Z+Y$ where $Z$ is a ...
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### Proving a distribution is not infinitely divisible

I'm trying to show the following: Show that the distribution on $\mathbb R$ with density $f(x) = \frac{1-\cos(x)}{\pi x^2}$ is not infinitely divisible. The characteristic function of this ...
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### Central limit theorem for two-sided Pareto distribution

I am trying to solve the following problem, which provides an example for a central limit theorem in spite of the fact that the variance is infinite. Consider the two-sided Pareto distribution with ...
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### Positivity condition of an integral from two functions

The density operator $\hat{\rho}$ representation by means of symplectic tomogram $\mathcal{W}(X|\mu,\nu)$ (that is a probability density function (pdf) of the quadrature $X$) is the inverse Radon ...
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### 2X has same distribution as sum of two iid copies of X under smooth transformation

Suppose $f:[0,+\infty)\rightarrow [0,+\infty)$ is a given strictly increasing smooth function starting from $f(0)=0$. I want to figure out under what conditions with respect to $f$ can we find ...
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### Characteristic function of the mixture of exponential and normal distributions

I've been calculating the characteristic function of a random variable with distribution $\frac{2}{3}E(\frac{1}{3}) + \frac{1}{3}N(6, 3)$. \begin{align} \phi(t) &= \int_{-\infty}^{0} e^{-itx}\frac{...
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### Is it true differentiability of characteristic function at 0 implies existence of the first moment.

The context is the following: Prove $|\cos(t)|$ is not a characteristic function of a distribution, while $\cos(t)$ is a characteristic function. I know you can prove it by inverse Fourier ...
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