# Tagged Questions

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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### Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
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### $X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0$and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$

$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic ...
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### Convergence of characteristic functions to $1$ on a neighborhood of $0$ and weak convergence

Prove the following statement: $X_n \Rightarrow 0$ (convergence in distribution) if and only if $(\exists\; \epsilon>0: |t|<\epsilon) \;\; \phi_n(t) \rightarrow 1$, where $\phi_n(t)$ is ...
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### For symmetric stable distributions, why is $\alpha \le 2$?

I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact. Suppose we are trying to come up with stable distributions. From the definition, ...
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### Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
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### A criterion for independence based on Characteristic function

Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be ...
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### Characteristic function of a standard normal random variable

The characteristic function of a random variable X is given by $$\Phi_X(\omega) = \mathbb{E}e^{j\omega X}=\int_{-\infty}^\infty e^{j\omega x}f_X(x) dx.$$ One can easily capture the similarity between ...
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### Continuous probability distribution with no first moment but the characteristic function is differentiable

I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere. Cauchy distributions do not ...
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### Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution

I am trying to derive Chi-square distribution. The random variale is $$U^2=\sum_{i=1}^k X_i^2$$ where $X$ is a random variable with normal standard distribution. What is the distribution of ...
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### Characteristic function of Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal ...
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### Proving that $\chi_{T^*}=\overline{\chi_T}$ and $m_{T^*}=\overline{m_T}$ (characteristic and minimal polynomials of adjoint map)?

For a linear map $T:V\to V$ where $V$ is a finite dimensional inner product space over $\mathbb{C}$, I know the result $\chi_{T^*}=\overline{\chi_T}$ (where $T^*$ is the adjoint map for $T$). My ...
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I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim ... 1answer 39 views ### Characteristic function using conjugate property To prove that e^{−i|x|} is not a characteristic function:$$e^{−i|x|} =\cos|x|-i \sin|x|.$$Its conjugate will be \cos|x|+i \sin|x| which is not equal to \phi(-x). Is my solution correct? 2answers 162 views ### Cantor Set : Characteristic function What is the characteristic function of the C Cantor set? I already found that the function is Integrable but i couldn't find the exact result of the charateristic function of the C Cantor set. 3answers 69 views ### Characteristic function of a square of normally distributed RV Let's assume that  X \sim \mathcal{N}(0,1) . I'm supposed to compute the characteristic function of  X^2 . As far as I got is that the density of  X^2  is  g(y) = \frac{1}{2\sqrt{2\pi}} ... 1answer 75 views ### Analytical continuation of moment generating function Let's say some distribution F(t) has finite moment generating function on an open ball (-R, R). M(x) = \sum m_n x^n /n! Let's extend M(x) to M(z) on a complex strip S = \{z| |Re(z)| ... 1answer 218 views ### Find characteristic function of random variable Let random variables X, Y, Z independent. X with uniform distribution on [-a,a], Y with Poisson distribution with parameter ν, Z with Bernoulli distribution with parameter p. Find the ... 1answer 134 views ### Characteristic function of a product of random variables I am facing the following problem. Let X,Y be independent random variables with standard normal distribution. Find the characteristic function of a variable  XY . I have found some information, ... 2answers 147 views ### Prove that X,Y are independent iff the characteristic function of (X,Y) equals the product of the characteristic functions of X and Y Let (\Omega,\mathcal A,\operatorname P) be a probability space X and Y be random variables on (\Omega,\mathcal A,\operatorname P) with values in \mathbb{R}^m and \mathbb{R}^n, ... 1answer 134 views ### Computing the characteristic function of a normal random vector The characteristic function of a random vector \boldsymbol{X} is \varphi_{\boldsymbol{X}}(\boldsymbol{t}) =E[e^{i\boldsymbol{t}'\boldsymbol{X}}]  Now suppose that \boldsymbol{X} \in ... 4answers 375 views ### Showing the expectation of the third moment of a sum = the sum of the expectation of the third moment Let X_{1},\cdots,X_{n} be independent, each with mean 0, and each with finite third moments. Show that E\left\{\left( \sum_{i=1}^{n}X_{i}\right)^{3}\right\} = \sum_{i=1}^{n}E\left\{ X_{i}^{3} ... 1answer 65 views ### Computing Conditional Characteristic Function I am trying to compute the characteristic function of the following: Let X and Y be random variables such that Y\mid X = x\sim N(0, x) with X\sim\mathrm{Po}(\lambda). Find the characteristic ... 2answers 535 views ### Characteristic function for positive part of random variables I need your help in solving the following problem: I have to calculate the characteristic function for the positive side of a random variable - simplest case: let Y be standard normal and Y^+ =\max ... 0answers 38 views ### Are the following functions characteristic functions of a random variable? Let be X an arbitrary random variable with characteristic function \varphi(t). Prove that$$\psi(t)=\frac{1}{2-\varphi(t)},$$and$$\chi(t)=\int_{0}^{\infty}\varphi(st)e^{-s}ds$$are also ... 0answers 287 views ### Characteristic function of compound Poisson process It is widely known that the characteristic function of a compound Poisson process is$$ \phi_X(u) = \exp \left(t\lambda \int_{\mathbb{R}} (e^{iux}-1) F(dx) \right). $$But if I try to derive it via ... 2answers 156 views ### Characteristic function of p(x) = \frac{1}{2} e^{-|x|}, -\infty < x < \infty Let X denote a real-valued random variable with an absolutely continuous distribution with density function p(x) = \frac{1}{2} e^{-|x|}, -\infty < x < \infty. Find the characteristic ... 1answer 88 views ### Showing \varphi(t)\neq 0 when \varphi is a characteristic function of an infinitely divisible distribution Let \varphi be a characteristic function of an infinitely divisible random variable. Show that \varphi(t) \neq 0 for all t. Sorry, I have no clue how to do it, because if the exponential is ... 1answer 277 views ### Characteristic function converges pointwise Is there any sequence of probability distributions such that their characteristic functions converge pointwise, but the sequence of prob. distributions itself does not converge weakly? :S 1answer 30 views ### Convergence of \chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}} for u_n \to u? Let u_n \to u in L^2(\Omega) and let I be an bounded interval. Does it follow that$$\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}} at least for a subsequence of $u_{n_j}$ ...
Suppose $E{|X|^{2}}<\infty$ and $E{X}=0$. Show that Var(X)$= \sigma^{2}<\infty$ (done), and that $\varphi_{X}(u)=1-\frac{1}{2}u^{2}\sigma^{2}+o(u^{2})$ (what I can't figure out how to find, ...
I have the following integral: $$f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt$$ where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...