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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4
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58 views

Does $|f\sin (x)|$ integrable on $\mathbb{R}$ imply that $|f|$ integrable on $\mathbb{R}$?

I guess not. Because we usually require $|f|$ to be integrable on ℝ so that it has the fourier transform. Can anyone give me an counterexample for the statement in the title? I have searched for ...
4
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3k 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
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0answers
68 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 ...
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313 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.
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85 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}\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 that ...
3
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96 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
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0answers
466 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
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23 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
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0answers
51 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
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47 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 $0< b < 1$. ...
2
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77 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
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64 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
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25 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
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199 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
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0answers
97 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 ...
2
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959 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 ...
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20 views

Characteristic function of basic affine process by rotation count algorithm

Hi everyone, I've had this frustating, silly problem for a while now. I've looked at the problem for a loooong time now, which may be one of the reasons I can't see the solution. I am trying to ...
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0answers
32 views

Some basic questions related to independence of random variables

I attend a lecture about Stochastic Processes even though I have not studied mathematics and some of the basics in probability theory are missing. So I hope you can help me with the following ...
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0answers
15 views

Characteristic function of an independent variable, does it involve complex values?

Let $$ x_k = \begin{cases} 1 & \mathrm{prob} (1/2)\\ -1 &\mathrm{prob} (1/2) \end{cases}$$ be independient random variables. Show that the characteristic function of the random variable ...
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24 views

Borel function and characteristic equation

Define a Borel probability measure $\mu_n $ by $\mu_n ({x}) = \frac{1}{n} $ for $x = 0, \frac{1}{n}, \frac{2}{n}, ..., 1-\frac{1}{n} $. Let $\eta$ be a Lebesgue measure on $[0,1]$. i) I'm to compute ...
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41 views

Characteristic function of truncated Cauchy distribution

Truncated Cauchy distribution (with a symmetric truncation $-X\le x\le X$) has the density $$f(x)=\frac{1}{2\arctan(X)}\frac{1}{1+x^2}$$ What is the characteristic function of this r.v.? Thank's! ...
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26 views

Transform Characteristic Function to distribution Function

I was wondering how we can calculate the distribution function of this characteristic function, $$C(t)= \frac{3+\cos t + \cos2t}{3}$$
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0answers
68 views

Clarification of Proof on Kac's Theorem for Characteristic Functions

There is a proof given here that I don't really understand, and was hoping someone more competent could explain it in some more detail: Moment generating functions/ Characteristic functions of $X,Y$ ...
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77 views

Recovering pmf from characteristic function

I'm having some trouble trying to recover the probability mass function of a discrete random variable from its characteristic function. I have seen that some continuous cases, you can recognize that ...
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0answers
47 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
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0answers
135 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 ...
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87 views

Probability:questions on characteristic functions

A well-known example to show that two random variables whose marginal distributions are normal, do not need necessarily be jointly normal is achieved by letting $X, Y $ have the following joint ...
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34 views

Integrating characteristic functions.

I need to find the radon transform of the following function. But I got stuck in finding this integral. Let $\chi$ be given by $$\chi(t) = \begin{cases} 1 & |t|< 1/2 \\ 0 & ...
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0answers
26 views

$X\sim\mathcal N(0,1)$, Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd?

If $X\sim\mathcal N(0,1)$ Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd ? ($\Phi_X^{(j)}(0):j^{th}$ derivative of the characteristic function of the r.v. $X$) We computed ...
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0answers
25 views

Finding Characteristic Exponents for $x^2 (x-1)^2 y'' + 4 (x-1)y' - 4x^2 y = 0$

I've found that the only regular singular point of this differential equation: $$x^2 (x-1)^2 y'' + 4 (x-1)y' - 4x^2 y = 0$$ is $x = 1$. How do I determine the characteristic exponents for it?
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26 views

Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
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0answers
30 views

Charateristic function evaluation

I have a signal given by the following equation: $$y_k = X_k S_0 + \sum_{l=0 \& l\neq k}^{N-1}S_{l-k}+n_k$$ where $X_k$ are independent and identically distributed random variables. $n_k$ is a ...
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0answers
79 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 ...
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49 views

I want to show $\phi_{X}(a_{1},a_{2},\cdots,a_{n})=\prod_{i=1}^{n}\phi_{X_{i}}(a_{i})$

Let $n \in \mathbb N$ and $X$ be an $\mathbb R^n$ valued random variable on $(\Omega ,\mathcal F,P)$ Define its characteristic function to be $$\phi_{X}(a)=E(e^{i\langle X,a\rangle})$$ where $a \in ...
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0answers
421 views

Characteristic Function of a Double Exponential (Laplace) Distribution

Let X have the double exponential (or Laplace) distribution with $\alpha =0$, $\beta = 1$: $f_{X}(x)=\frac{1}{2}e^{-|x|}$, $-\infty < x < \infty$. Show that $\varphi _{X}(u)=\frac{1}{1+u^{2}}$. ...
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383 views

Characteristic functions in set theory

The book I am studying has the definition of a characteristic function as follows. Let $A\subseteq{X}$. Then $$\chi_A(x) = \begin{cases} 1, & \text{if $x\in{A}$} \\ 0, & \text{if ...
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0answers
74 views

Series expansion at $0$

Given: $X$, $Y$ iid random variables, $\mathbb E(X) = 0$, $\mathbb E(X^2) = 1$; $X+Y$ and $X-Y$ are independent; $\phi$ is the characteristic function of $X$ and $Y$ and $ \psi: t \rightarrow ...
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0answers
127 views

Probability density function of $A = B + C$ via Joint Characteristic function of $B$ and $C$

This problem is actually a subproblem of a longer derivation that I am trying to understand. I hope that I striped away all the unnecessary stuff that is not relevant. Please correct me if the ...
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198 views

Questions about characteristic functions and continuity

If there is a subset $S$ in $\mathbb{R}^n$ consider the characteristic function $\chi_S: \mathbb{R}^n \to \mathbb{R}$. i) what would be the value of $\chi_S(p)$ for any $p \in \mathbb{R}^n$? Attempt ...
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0answers
14 views

Characteristic function of an asymmetric Laplace distributed random variable

What is the characteristic function of a random variable with density $$f_X(x) = \frac{1}{2} [ 1_{x>0} \, a e^{-a x} + 1_{x<0} \, b e^{b x} ], \; \; \; \quad a,b > 0 \quad \quad ? $$ My ...
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18 views

Check if a distribution is discrete or continuous from the characteristic function of the distrution?

Is it possible to check if a distribution is discrete or continuous from the characteristic function/Laplace transform of a distribution?
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13 views

Problem in the correspondence between boolean rings and boolean algebra through characteristic functions

I was working on the relation between boolean algebras and boolean ring and that they are in fact, the same object. But I find something which seems to be incorrect, It's quite long and I try to give ...
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0answers
26 views

Probability - Characterizing goodness of moment matching method.

I have a question about how to characterize the goodness of approximating a distribution using its moments. Suppose I have a probability density function $p(x)$ (e.g., normal distribution), and I am ...
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0answers
31 views

Solution to this non linear Equation

Hi all I have this equation $$ (y-\beta+2\beta e^{yd})(y+\beta -2 \beta e^{-yd})+\beta^2=0 $$ Solving for y, I have been told that the solution is y = 0 repeated roots. I can see why (from ...
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0answers
44 views

Integral of Normal Distribution with imaginary unit

Hi I need some help with the following integral. $$ \int_{-\infty}^{\infty} \operatorname{e}^{itx} \cdot \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \operatorname{e}^{\frac{-(x - \mu)^2}{2\sigma^2}} \mathrm ...
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0answers
35 views

Measure of $\chi_\mathbb{Q}(x)$?

$\chi_\mathbb{Q}(x) = 1$ if $x \in \mathbb{Q}, 0$ otherwise. Well $\chi_\mathbb{Q}(x)$ is a measurable function if $\mathbb{Q}$ is a measurable set. $\mathbb{Q}$ is a measuable set under the Borel ...
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0answers
87 views

Characteristic function of a exponential random variable, problems with complex integral.

I tried to compute the characteristic function of a random variable, which is exponential distributed with parameter $\lambda$: \begin{align*} \varphi(t) &= \mathbb E[e^{itX}] = ...
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57 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 ...
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38 views

Limit of the expectation of the sum

Show that for $g(t)= E \left\{\sum_{n=3}^{\infty}\frac{(iut)^{n}}{n!}\right\}$ that $\lim_{t \to 0} \frac{|g(t)|}{t} =0$. I think I should bound it and then use LDCT, but I'm having trouble doing ...
0
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0answers
52 views

Find the characteristic equation of a recursive function

I want to determine whether the following recursive function is unstable; $$ x(t+1) = \left( wx+sx(t)^b \over w+x(t)^bs + (1-x(t))^b(d-s) \right) $$ Wikipedia is telling me that I want to have the ...