Articles of characteristic functions

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 $\boldsymbol{X} \in N(\boldsymbol{\mu},\boldsymbol{\Lambda})$. We observe that $Z = \boldsymbol{t}’\boldsymbol{X}$ has a one dimensional normal distribution. the parameters are $m = E[Z] = \boldsymbol{t}’\boldsymbol{\mu}$ and $\sigma^2 = \mathrm{Var}[Z] = \boldsymbol{t}’\boldsymbol{\Lambda}\boldsymbol{t} $($\boldsymbol{\Lambda} $ is the covariance matrix). since $\textbf{(1)}\enspace \varphi_{\boldsymbol{X}}(\boldsymbol{t})=\varphi_{z}(1) = \exp\{im […]

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 $E\left\{\left( \sum_{i=1}^{n}X_{i}\right)^{3}\right\} = \sum_{i=1}^{n}E\left\{ X_{i}^{3} \right \}$. Thanks in advance!

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, especially the fact that if $ \phi_X,\phi_Y $ denote characteristic functions, then $$ \phi_{XY}(t) = \mathbb{E}\phi_X(tY).$$ The only problem is that the proof required […]

Find characteristic function of ZX+(1-Z)Y with X uniform, Y Poisson and Z Bernoulli

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 characteristic function of random variable $Z*X+(1-Z)Y$ I tried use the this idea, but i don’t now, how i can apply it for discrete distribution. Why are […]

Step Function and Simple Functions

Definitions: Simple Function: Any functions that can written in the form:$$s(x)=\sum_{k=1}^na_n\chi_{A_n}(x).$$ Note the finite terms here. It should follow that neither all simple functions are step functions, nor all step functions are simple function. e.g. Would not Cantor Function or Devil’s Staircase be example of step function but not simple (note again the finite)? I […]

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 one show that it is a characteristic function? (Typed this on my phone – my apologies if there’s poor formatting)

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 $S = \{z| |Re(z)| <R\}$. I need to prove that $M(z)$ is also analytic on $S$(thus it is an analytic continuation). What is the […]

Is $t\mapsto \left|\cos (t)\right|$ a characteristic function?

Can anyone explain how I can prove that either $\phi(t) = \left|\cos (t)\right|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.

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 fulfill this, as their characteristic functions are not differentiable at $0$. Does anybody here have an example?

Reducing an indicator function summation into a simpler form.

Context I am attempting to reduce the space I need to store in an array in a program. I have made it so that the indices are always sorted. There are no indices where they are equal, and no indices where the $n$th index is greater than any of the previous indices. I also eliminated […]