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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0answers
70 views

A random variable is symmetric if and only if its characteristic function is real-valued

Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric. Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega \...
5
votes
0answers
69 views

Properties of characteristic functions under statistical dependence

Given random variables $X,Y,Z$,and $\phi(.)$ denoting the characteristic function, I can see that the following is true when $Z$ is independent of $X,Y$: $|\phi_{X+Z,Y} (t, s) − \phi_{X+Z}(t)f_{Y} (s)|...
4
votes
0answers
42 views

Finding the odometer function of an abelian sandpile

As a computer scientist and "armchair" mathematician, I'm trying to replicate the images found here of Abelian sandpiles on a square lattice, where the initial configuration is $n$ chips on a single ...
4
votes
0answers
56 views

Exsistence of random variable $U[-1,1]$

I have problems with the following question: Let X be the random variable such that $X\sim U[-1,1]$. Does there exsist a random variable Y, independent to X, such that $X+Y\sim 2Y$? I thought that ...
4
votes
0answers
124 views

Moment generation function -> characteristic function uniqueness

Here's my proof that moment generation function (if exists) uniquely determines characteristic function. Can you please see how to make it more rigorous or improve in either way (e.g. by citing ...
4
votes
0answers
61 views

Holomorphic extensions of characteristic functions

Let $\chi(\xi) := \mathbb{E}e^{i \xi X}$, $\xi \in \mathbb{R}^d$, be the characteristic function of a random variable $X$. It is widely known that $\chi$ admits a holomorphic extension to the strip $\{...
4
votes
0answers
79 views

Uniform convergence of characteristic functions implies uniform convergence of distribution

Let $F(x)$ and $(F_{n})_{n\geq 1}$ be some distribution functions and let $\varphi(t)$ and $(\varphi_{n})_{n\geq 1}$ be their respective characteristic functions. I am trying to show that if: $\sup_t ...
4
votes
0answers
133 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}$ are i.i.d. $\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows ...
4
votes
0answers
2k views

Continuity of a Characteristic function

Let $A$ be a subset of $\mathbb{R}^n$. Show that the characteristic function $\chi_A$ is continuous on the interior of $A$ and on $A^c$ but discontinuous on the boundary of $A$. My attempt: Suppose ...
4
votes
0answers
5k views

Characteristic functions of random variables (Poisson, Gamma, etc.)

My self-study in measure and probability theory as finally brought me to the subject of characteristic functions, and I have not handled these in the past with any rigor at all, so all of this is ...
3
votes
0answers
43 views

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 ...
3
votes
0answers
29 views

Can inversion integral of characteristic functions on a finte interval be bounded?

For a real-valued uni-variate r.v. $X$, with pdf $f(x)$ and absolute integrable cf $\varphi(t)$, we have the following transform:$$2\pi f(x)=\int_{-\infty}^{\infty}e^{-itx}\varphi(t)\,dt.$$ However, I ...
3
votes
0answers
97 views

Invert a somewhat tricky characteristic function to find density function

I am interested in find the probability density function corresponding to the characteristic function $\phi(t) = \left(\frac{1 - i b t}{1 - i t}\right)^c$ where $c > 1$ and and $0< b < 1$. ...
3
votes
0answers
112 views

An inequality with a characteristic function

It's my first question here, hi. In fact, it derives from my probability theory homework, which appears to be unusually difficult (or I just don't see something): Suppose $X$ is a real valued random ...
3
votes
0answers
114 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 ...
3
votes
0answers
474 views

Why the probability characteristic function is always exist but moment generation function is not always exist?

I know that the characteristic function is always exist and moment generation function is not always exist instantly but don't know exactly mathematically.
3
votes
0answers
126 views

Integral of the Normal Characteristic Function

The characteristic function of the $N$-variate Normal distribution is $$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ }...
3
votes
0answers
544 views

Distribution of the sum of iid Beta-Negative-Binomial random variables

I am facing a problem when trying to calculate the distribution of the sum of iid Beta-Negative-Binomial random variables or for that matter if only parameter $r$ is different. To get a hint to how ...
2
votes
0answers
26 views

$e^{\varphi -1}$ characteristic function

So I am trying to figure out whether $e^{\varphi-1}$ is a characteristic function given that $\varphi$ is. I know that linear combinations of characteristic functions and the real part of a ...
2
votes
0answers
33 views

How to prove convergence for the following?

Let $X_m$ be distributed as follows: $$\mathbb P(X_m=\pm m)=\frac{1}{2m^\beta}, \mathbb P(X_m=0)=\frac{m^\beta-1}{m^\beta}.$$ Let be $S_n=\sum_{m=1}^{n}X_m$, then I. If $\beta>1$, then $\...
2
votes
0answers
41 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 ...
2
votes
0answers
73 views

Convergent sequence of characteristic functions with continuous integrable limit.

Suppose $\varphi_{n}$ is a sequence of characteristic functions converging pointwise to a function $\varphi$ which is continuous and absolutely integrable. I want to show that $\varphi$ is the ...
2
votes
0answers
59 views

Proofs of some characteristic function properties

I saw these properties in the wikipedia page but I was unable to prove them. I had an idea to construct a random variable with the desired characteristic function, yet I haven't managed to that so far....
2
votes
0answers
34 views

Characteristic function of $\chi^2$ distribution with $n$ degrees of freedom

I'm computing the formula for the characteristic function of the random variable $X \sim \chi^2(n), $ $n\in\mathbb{N}$. After some substitutions in the integral and some messing around with certain ...
2
votes
0answers
52 views

Distribution of $aX+bX^2+cX^3$ where $X$ is standard normal

I am looking for some distributional characteristic (for example a characteristic function) of a random variable which is defined as $aX+bX^2+cX^3$, where $X$ is a standard normal variable. Is there ...
2
votes
0answers
70 views

Techniques in analytic number theory

I'm fairly new to the subject and trying to figure things out. Would be nice to hear some ideas and trickery for what follows. Suppose we wish to show there exists an integer $x$ in some finite set $...
2
votes
0answers
33 views

Can anyone shed some light on the random variable which has the following characteristic function?

I have a random variable whose characteristic function is of the form \begin{equation} \mathbb{E}[e^{itX}] = \frac{(1-it)^a}{(1-2it)^{\frac{a}{2}}}\,, \end{equation} where $0<a<1$ I am not ...
2
votes
0answers
49 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
2
votes
0answers
317 views

Derivation of Gamma distribution characteristic function reference?

I was wondering if there was a derivation of the Gamma distribution characteristic function without expanding the $e^{itx}$ into an infinite summation?
2
votes
0answers
46 views

Joint characteristic function of $X$ and $F(X)$

$X$ is a random variable. Its distribution function and characteristic function are $F_X$ and $\phi_X$, respectively. Then, we know, $F_X(X)$ follows uniform distribution. Let's say, $U=F_X(X)$. My ...
2
votes
0answers
127 views

Characteristic function of complex valued random variable

1) How is Characteristic function of a complex valued random variable defined? Should it be considered as vector of real random variables or the definition in wiki be used? 2) Also how is the ...
2
votes
0answers
64 views

How to use Duhamel's principle to solve wave equation

Let $x\in\mathbb{R}$,$t>0$, $$\frac{\partial u}{\partial t}+5\frac{\partial u}{\partial x}=e^{-t}\sin x,$$ $$u(x,0)=(1-x^2)_{+}.$$ Using Duhamel's principle to solve it. By Duhamel's principle, ...
2
votes
0answers
35 views

Characteristics function and moments of multivariates

I have been reading this paper recently-- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.199.2157&rep=rep1&type=pdf This is a paper by Nengjiu Ju who uses talyor series to express ...
2
votes
0answers
83 views

$e^{-d|z|^\alpha}$, $d\geq0,0<\alpha\leq2$, is characteristic function of a stable distribution

Problem: Prove that $e^{-d|z|^\alpha}$ is characteristic function of a stable distribution, if $d\geq0$ and $0<\alpha\leq2$. A note on the definition of stable: Note that a measure $\mu$ ...
2
votes
0answers
117 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$? Let $\varphi_X(t)$ be ...
2
votes
0answers
193 views

Characteristic functions and conditional distributions?

Say X and Y are random variables and we're interested in the conditional distribution of X given Y, can we make this calculation using only characteristic functions in a straightforward manner? If so ...
2
votes
0answers
26 views

Where have I used the assumption that $X\in L_2$?

Let $X\in L_2$ be a random variable and $g$ a positive real function. Let $I$ be an interval and $b>0$, and suppose that $\forall x\in I\ g(x)>b$. I have to show that: $$\operatorname E(g(X))\...
2
votes
0answers
70 views

Order Statistics interval sizes

Suppose an i.i.d. sample of size $n \geq 2$ drawn from a known distribution with density $g$. Let us note the associated order statistics as $(X_{(1)}, \ldots ,X_{(n)})$. I am interested in the number ...
2
votes
0answers
202 views

properties of characteristic function

Let $X,Y$ be two independent random variables having the same distribution, centred and with variance 1, $\phi$ is the characteristic function of $X$ and $Y$. If $X+Y$ and $X-Y$ are independent, show ...
2
votes
0answers
290 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 ...
2
votes
0answers
91 views

Does the Riemann zeta function tell us about the order theoretic properties of the natural numbers?

The classical Möbius function $\mu(n)$ fulfills the multiplicative inversion formula, e.g. see this thread. Now I see in the theory of posets, they generalize the concept of that function, see ...
1
vote
0answers
13 views

How to compute the $PDF_{X}(x)$ of $X$ if it cannot be Fourier inverted from the characteristic function $CF_{X}(z)$?

I have a positive random variable $X>0$. I have to compute the probability density function $$PDF_{X}(x)$$ I can compute in closed-form the extended characteristic function ($z \in \mathbb{C}$) $$ ...
1
vote
0answers
40 views

Characteristic function of a lattice distributed random variable

Let $X$ be a random variable. $X$ is called lattice distributed if there exist real numbers $a, b$ such that $P(X \in a +b\mathbb{Z})=1$. Show that $X$ is lattice distributed if there exists $v\neq 0$ ...
1
vote
0answers
14 views

Characteristic function of triangular distribution over $[0,2]$.

Here it is shown that if $X_1,\dots ,X_n$ are iid random variables, then the characteristic function of $S=\sum_{i=1}^nX_i$ is the product of the respective characteristic functions of the $X_i$. ...
1
vote
0answers
30 views

Difference between the characteristic function of a sum of independent vectors and a Gaussian

In the book "Sums of Independent Random Variables" by Petrov the following lemma appears in page 109, in preparation to prove the Berry-Esseen inequality Let $X_1,...,X_n$ be independent random ...
1
vote
0answers
25 views

characteristic method for wave equation in a non-uniform string

wave equation in a non-uniform is $u_{tt}=c(x)u_{xx}$, $c(x)=1$, when $0<x<1/2$, $c(x)=2$, when $1/2<x<1$, Does $u(x,t)$ has the form like $u(x,t)=f(x-t)+g(x+t)$ when $0<x<1/2$, $...
1
vote
0answers
52 views

Solve IVP with the method of characteristics (quasilinear PDE) (+shockwaves)

The question i just can't figure out one bit is: Solve for the continious function $u(x,t)$: $$u_t+(1-2u)u_x=0 $$ $$-\infty < x < \infty, t>0$$ $$u(x,0)= \left\{ \begin{matrix} \frac{1}{4} &...
1
vote
0answers
54 views

Suppose that $Y_{\lambda}=^{d}P(\lambda)$. Prove that $[Y_{\lambda}-\lambda]/{\sqrt{\lambda}}\to^{d}N(0,1)$

Suppose that $Y_{\lambda}=^{d}P(\lambda)$. Prove that $[Y_{\lambda}-\lambda]/{\sqrt{\lambda}}\to^{d}N(0,1)$ when $\lambda \to \infty$ using characteristic functions. So $$\phi(t)=\sum_{k=0}^{\infty} \...
1
vote
0answers
54 views

integral of $e^{-\imath x}$ from $a$ to $\infty$ and characteristic functions

In the following result: $$\frac{1}{2 \pi} \int_{-\infty}^{\infty} \Phi(u) \left[\int_a^{\infty} e^{-i u x}dx\right] du = \frac{1}{2} + \frac{1}{ \pi} \int_0^{\infty} \Re\left[\frac{e^{-i u a} \Phi(u)}...
1
vote
0answers
26 views

PDE -how to find characteristic function

How to find characteristic of the following equation $\mathbf{e^{-x}u_{xx}+2 \hspace{1pt} e^yu_{xy}+e^x \hspace{1pt} u_{x}=0}$ would the characteristic derived from the below equation? $e^{-x}(\...