0
votes
1answer
39 views

Characteristic function of logarithm of random variable

If I know the characteristic function $\phi_X(t)$ of a random variable $X>0$, how can I write the characteristic function $\phi_Y(t)$ of $Y=\log(X)$? I know that $\phi_X(t)=E[e^{itX}]$ and ...
1
vote
1answer
62 views

Characteristic Function Inversion

I am studying the relationship / bijection between characteristic functions and CDFs. In particular, given a characteristic function $\phi$ it is posible to recover the cumulative density function ...
0
votes
1answer
159 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
0answers
51 views

Possible values of characteristic functions (Fourier transforms)

Can a characteristic function $\varphi_{X}(u)$ from probability theory (the Fourier transform of a probability measure) ever equal zero for either any value of $x$ or any value of $u$? This has been ...
2
votes
0answers
87 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...
1
vote
1answer
161 views

Inverse Fourier transform of characteristic function

Let $Q$ be a measurable set in $\mathbb{R}^2$ Let \begin{equation} 1_Q(\textbf{x}) = \left\{ \begin{array}{ll} 1 & \mbox{if $\textbf{x} \in Q$},\\ 0 & \mbox{otherwise},\end{array} \right. ...
0
votes
1answer
129 views

How do you invert a characteristic function, when integral does not converge?

I need to find the probability density of some distribution with characteristic function given by: $$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$ I know the formula for inverting a ...
2
votes
2answers
89 views

Uniqueness of distribution with moments $M_n$ if $\limsup_{n\to\infty} \frac{1}{n}\sqrt[n]{M_n}$ finite

There's a theorem which states that the moments, i.e. $M_n = \mathbb{E}\left(X^n\right)$, of a distribution uniquely identify the distribution if $$ R := \left(\limsup_{n\to\infty} ...
1
vote
1answer
66 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 ...
3
votes
2answers
331 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 ...
1
vote
3answers
291 views

Is this function - characteristic function of a random variable?

$\phi(t)= \begin{cases} 1,&\text{if $|x|< 1$;}\\ e^{-(|x|-1)^2} ,&\text{if $|x|\geq1$.} \end{cases} $ Can anyone help?
5
votes
1answer
252 views

Determining if something is a characteristic function

Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$ Question: Given a function, say $\psi(t)$, how does ...
2
votes
1answer
108 views

Fourier transform of characteristic function in a sphere

A similar question was asked before for an interval in $\mathbb{R}$. I wonder how to do it for a characteristic function of $\{x\in\mathbb{R}^3:|x|<r\}$ i.e. I want to calculate $$ ...
5
votes
1answer
398 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, ...