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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.

Given a set $A \subseteq X$, the characteristic function of $A$ is the function $\chi_A : X \to \mathbb{R}$ given by

$$\chi_A(x) = \begin{cases} 1 & x \in A\\ 0 & x \notin A. \end{cases}$$

Characteristic function defined as above is a synonym for indicator function.

1252 questions
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How to show the following characteristic function is positive definite

Let $\phi$ is the characteristic function of a probability measure on $\mathbb{R}$, how can I prove $\sum_{i,j=1}^{n}{\phi(t_i-t_j)\bar{\phi}(t_i-t_j)\xi_i\bar{\xi_j}}\geq 0$ for all $n\in\mathbb{N}$, $\xi_1,\ldots,\xi_n\in\mathbb{C}$, and…
Jack
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Why does the characteristic function always exist?

I've read that the characteristic function of a probability distribution always exists because it's bounded. However, the characteristic function is still Taylor expanded in terms of the moments of a given probability distribution. Given the the…
Dargscisyhp
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the characteristic function of Levy Distribution

The standard Levy distribution has the PDF: $$f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2x}}\frac{1}{x^{3/2}},$$ where $x\geq0$. My question is how to compute its characteristic function: $$ \phi(t)=\int_{-\infty}^{\infty} e^{jtx}f(x)\mathrm{d}x. $$ I…
user129252
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How to calculate $\int_{0}^{\infty} \frac{e^{itx}}{(1+x^{c})^{k+1}} \, \mathrm{d}x $?

I was calculating a characteristic function and I couldn't compute this integral: $$\int_{0}^{\infty} \frac{e^{itx}}{(1+x^{c})^{k+1}} \, \mathrm{d}x $$ Where $k,c>0$, any hint would be of great help.
Toney Shields
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Why is $|\varphi(t)|$ not necessarily a characteristic function?

I came across the following statement in a book: If $\varphi(t)$ is a characteristic function, then $|\varphi(t)|$ is not necessarily a characteristic function. Here's my argument: By Bochner’s theorem, we can check $|\varphi(t)|$ satisfies the…
1024
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show that $\exp(-t^4)$ is not a characteristic function

I found, by assuming that exists a random variable $X$ that accepts $\exp(-t^4)$ as characteristic function , $E[X^2] = 0$, which means that $P[X=0] = 1$ implying the function is $0$. It's acceptable to use this fact to prove what I want, but I…
newtonplusapple
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Why can I use the Riemann-integral here?

Let $Z\sim\mathcal{N}(0,1)$ (i.e. a random variable which distribution is the standard normal distribution). Determine the characteristical function of $Z$. It is $\mathbb{P}_Z=f\lambda$ with $f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}$ and…
mathfemi
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Characteristic function of $\frac{1-\cos(x)}{x^2}$.

I am trying to compute the characteristic function of the density $f(x)=\frac{1-\cos(x)}{x^2}$. But I do not know how to do it, I was trying to use the residue theorem to compute $$\phi(t)=\int_{-\infty}^\infty e^{itx}\frac{1-\cos(x)}{x^2}dx$$ but…
Don P.
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characteristic function properties

In lecture, we had the following corollary (without proof, unfortunately): If $ A \in (0,2) $ and $X$ is a random variable (real-valued) with the following characteristic: $$ \mathbb P(X > x) = \mathbb P(X < -x) = \frac{x^{-A}}{2} \text{ for any } x…
JohnD
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How to find the subset of a characteristic function?

For example, given the characteristic function XA + XB - XA $ \cap $ B where A and B and subsets of set S. How to find the subset of the characteristic function?
Madno
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Given a characteristic function, find the distribution.

I'm new to characteristic functions and I would really appriciate some help with the following question: "Give the distribution which has characteristic function $\varphi(t)=cos(t)$." I've tried to go backwards by the definition of a characteristic…
AnnieFrannie
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Show that $\int_{0}^{1} \phi(yt)\, dy$ is a characteristic function, where $\phi(t)$ is itself a characteristic function.

I tried: \begin{align}\int_{0}^{1} \phi(yt)\, dy &= \int_{0}^{1} E(e^{iytX})\, dy\\ &= \int_{0}^{1} E(\cos ytX + i\sin ytX)\, dy\\ &= \int_{0}^{1} E(\cos ytX)\, dy + i\int_{0}^{1} E(\sin ytX)\, dy\\ &= \text{??} \end{align} No idea where to go from…
Forklift17
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Characteristic function problem

first time poster so be nice! Here's the problem: Let $\phi(t)$ be a characteristic function, then $e^{\lambda(\phi(t)-1)}$ is a characteristic function. Pretty stuck, any help appreciated!
Sebastian
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Finding the characteristic function

Let ${\{X_i \} }$ $i≥1$ be iid random variables with distribution $$P(X_i = 1) = P(X_i = -1) = 1/2$$ Define $Z_n = \sum_{i=1}^n \frac{X_i}{2^i}$ and let $Z = \lim_{n \to \infty} Z_n$ Find the characteristic function of $Z_n$ and…
thedumbkid
  • 661
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Characteristic function, convergence in distribution

Problem: Let $X_{1},\ X_{2},\ \cdots$ be independent and identically distributed (i.i.d.) random variables having the ch.f. of the term $1-c|t|^{a}+o(|t|^{a})$ as $t\to0$, where $0
sukc
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