# 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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### Sufficient condition for the characteristic function of a discrete distribution to be decreasing

Let $X$ be a discrete random variable defined on the lattice such that $P(X \in \mathbb{Z}/h) = 1$, where $\mathbb{Z}/h$ represents the set of integers scaled by a factor of $h$. I wonder if there ...
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### Showing weak convergence of dirac measure using characteristic functions [closed]

I want to show if $\mu_n=\frac{1}{n}\delta_n+(1-\frac{1}{n})\delta_0$ converges weakly using characteristic functions. I know that the characteristic function of $\delta_0$ is the constant function $1$...
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### Inversion formula for characteristic functions whose argument is complex with non-zero real part

In this paper they use an inversion formula for characteristic functions whose argument is complex with non-zero real part. Namely, given a random variable $X$ taking values in $(0,\infty)$, they ...
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### PDF or CDF of product of two independant random variables

I'm looking for a formula for computing the PDF (I would prefer the CDF if possible) of the product of two independent random variables. I found the Mellin integral/transform is the analytical ...
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### Why is $\lim_{n\to \infty} n\chi_{(0,\frac{1}{n})} = 0$? [closed]

Why is $\lim_{n\to \infty} n\chi_{(0,\frac{1}{n})} = 0$? Can you provide a short (trivial) explanation? The sequence in the limit only have two possible values: $0$ and $n (\to \infty)$ so it is a ...
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### Can $\mathbb{R}$ be expressed as a disjoint union of two dense locally non-zero measure measurable subsets?

I was trying to prove that piecewise-constant functions on $\mathbb{R}$ are not dense in $L^\infty(\mathbb{R})$ and I was looking for a $L^\infty$ function that cannot be "approximated" with ...
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### Exercise 2.3.29 from Stroock's "Probability Theory: an Analytic View"

I am working on Exercise 2.3.29 from Stroock's "Probability Theory: an Analytic View". Let $X$ be a multivariate normal random variable defined on a probability space $(\Omega, F, P)$ with ...
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### On the charateristic function of random variables

For all random variables that admit a probability density function (PDF), their characteristic function provides an alternative way to completely define its probability distribution. Why is that? The ...
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### Express Random Variable as Sum of Independent Random Variables from Characteristic Function

I am presented with the following Characteristic Function $$\phi(\xi) = e^{-2\vert \xi \vert^{1.5} - 0.3 \xi^2 + 1.5i\xi}$$ corresponding to some random variable, and I am tasked with finding the ...
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### Confused with Characteristic Function as Fourier Transform of Density Function

The characteristic function of a random variable $X$ is defined as the expectation of the function $e^{itX(\omega)}$ i.e. $$\int e^{itX(x)}\rho(x)\,dx$$ where $\rho$ is the probability density. How is ...
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### Should the characteristic function of a probability distribtion decay to zero?

I am a theoretical quantum physicist trying to find probability distributions for heat transfer in an open quantum system. I am testing an approximate method with an analytically exact one, so I know ...
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### $X$ with $\mathbb P(X=1)=\mathbb P(X=-1)=\frac12$ is not infinitely divisible

I want to show that $X$ with $\mathbb P(X=1)=\mathbb P(X=-1)=\frac12$ is not infinitely divisible. This means I need to show that there $\exists n\in\mathbb N$ such that $\nexists X_1,\dots,X_n$ i.i....
I am missing a step in the proof of Theorem 15.22 of Probabiliy Theory by A. Klenke (3rd version). The theorem states that, given a tight family of probability measure on $\mathbb{R}$, the family of ...