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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7
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1answer
1k views

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 $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
1
vote
1answer
491 views

$X$ 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 ...
2
votes
1answer
495 views

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 ...
7
votes
1answer
688 views

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, ...
4
votes
2answers
1k views

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 ...
5
votes
2answers
1k views

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 ...
14
votes
1answer
6k views

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 ...
6
votes
1answer
369 views

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 ...
2
votes
1answer
174 views

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 ...
1
vote
2answers
315 views

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 ...
1
vote
3answers
57 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}} ...
1
vote
1answer
929 views

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 ...
0
votes
1answer
596 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

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 ...
1
vote
1answer
38 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?
1
vote
2answers
107 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.
2
votes
1answer
116 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, ...
3
votes
1answer
203 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 ...
2
votes
0answers
272 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 ...
1
vote
1answer
111 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 ...
1
vote
4answers
324 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} ...
1
vote
1answer
271 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
0
votes
1answer
51 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)| ...
3
votes
1answer
60 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 ...
3
votes
2answers
500 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 ...
2
votes
2answers
154 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 ...
2
votes
1answer
87 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 ...
2
votes
1answer
235 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
1
vote
2answers
114 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 $\mathbb{R}^n$, ...
1
vote
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 $u_{n_j}$ ...
1
vote
1answer
139 views

Characteristic function of an r.v. with finite variance and zero mean.

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, ...