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 views

How to use Duhamel's principle to solve wave equation

Let $x\in\mathbb{R}$,$t>0$, $$\frac{\partial u}{\partial t}+5\frac{\partial u}{\partial x}=e^{-t}sinx,$$ $$u(x,0)=(1-x^2)_{+}.$$ Using Duhamel's principle to solve it. I know how to use Duhamel's ...
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0answers
14 views

Differential equation whose solution is Erlang distribution

I am working on a proof (Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)) and got stuck. Now I would like try ...
0
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2answers
34 views

Solving characteristic equations…

Can someone explain to me what my teacher is doing? $x^2 - ax - b = 0$ ..? Isn;t he using the quadratic formula to solve this problem? If that's the case, then where is the $c$ at? Shouldn't he have ...
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0answers
29 views

Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)

The random variables X and Y have the following properties: X is positive, i.e., $P\{X > 0\} = 1$, with continuous density function $f_X(x)$, and $Y\mid X$ has a uniform distribution on $\{0,X\}$. ...
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1answer
16 views

Semi-Linear First Order PDE (with non-linear reaction term)

I've been trying to figure out this problem for a while and was wondering what you thought. I have the PDE: $\partial c \over \partial t$ + $K \over r^2$ $\partial c \over \partial r$ + $Da \ c \over ...
2
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1answer
31 views

show that $\exp(-t^4)$ is not a characteristic function

I found, by assuming that exists a random variable $X$ that accepts $\exp(-t^4)$ as characteristic function , $E[X^2] = 0$, which means that $P[X=0] = 1$ implying the function is $0$. It's acceptable ...
4
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1answer
84 views

Is there any short proof of this classical problem?

Let $X,Y$ be two i.i.d. r.v.'s with zero mean and unit variance. If $X+Y$ and $X-Y$ are independent, then $X$ and $Y$ are both standard normal distributed. Is there any short proof for this problem?
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1answer
26 views

Characteristic Function limit to 0

When calculating the limit of the following characteristic function $$ \frac{1}{n+1}\left[ \frac{1-\exp\left( \left(n+2 \right)it \right)}{ 1-\exp(it) } \right]$$ and taking its limit when ...
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1answer
29 views

Fourier transform of indicator function

Given a set of complex numbers $\mathcal A$, is there a convenient solution for the Fourier transform of its indicator function $\chi_{\mathcal A}(z)$? More specifically, if $\mathcal A$ is a set of ...
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1answer
15 views

Binomial-Poisson limit

I want to show that if $Z_n$ has the binomial distribution with parameters $n$ and $\lambda/n$ with $\lambda$ fixed, then $Z_n $ converges in distribution to the Poisson distribution, parameter ...
2
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1answer
37 views

calculating characteristic polynomial in $\mathbb{R}^n$

Given some hyperplane arrangement $\mathcal{A}$, we call any subset $\mathcal{B}\subseteq \mathcal{A}$ $\textit{central}$ if $$\displaystyle \bigcap_{H\in \mathcal{B}}H\neq \emptyset.$$ There is a ...
3
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2answers
28 views

How to use Cayley-Hamiltonian theorem in proving upper bound on linear space $W$?

If $W = span(I,A,A^1,A^2, \dots)$. What is the upper bound on dimension of $W$? All matrices are $n \times n$. I know that the dim($W$) $\leq n$, by the Cayley-Hamiltonian theorem. However, I don't ...
2
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1answer
21 views

Characteristic function of $\sum_{t=1}^N a_t X_t$ given certain independence conditions

Let $S=\sum_{t=1}^N a_tX_t$ where each $a_t$ is Bernoulli with probability $\frac{1}{2}$ for $1$ and also $\frac{1}{2}$ for $0$. Moreover, it is also given that the vector $(a_1,\ldots,a_N)$ is ...
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0answers
16 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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1answer
35 views

New characteristic function from old

The question I want to do says: Let $f(u,t) : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function, such that for each $u$, $f(u, \cdot)$ is a characteristic function, and such that for each $t$, ...
0
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1answer
18 views

Independence of a random variable and a sub-$\sigma$-algebra

I am having trouble understanding one of the steps in the proof of the following lemma. Let $X$ be a random $d$-vector and $\mathcal{A}$ a sub-$\sigma$-algebra on the probability space ...
2
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0answers
24 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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1answer
28 views

New characteristic functions from old

I am doing an exercise which says: If $f$ is a characteristic function, then show that $$ F(t):= \int_0^{\infty} f(ut)e^{-u}du $$ is again a characteristic function. Is this answer correct? Let ...
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1answer
22 views

Proving $\chi_{f^{-1}(A)}(i) =(\chi_{A} \circ f)(i)$

Does this work for any $f$? \begin{equation} \chi_{f^{-1}(A)}(i) = \begin{cases} 1, & \text{if $i \in f^{-1}(A)$} \\ 0, & \text{if $i \not\in f^{-1}(A)$} \end{cases} = \begin{cases} ...
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1answer
32 views

Integrals of indicator functions question

I have a result $\int_X \int_Y \mathbb{1}[h(x,y) < \mu]dP(y)dP(x) < a$ and I am trying to resolve the integral $\int_X \int_Y \mathbb{1}[|f(x) - g(x)| > \frac{\mu}{2}] \mathbb{1}[h(x,y) ...
3
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1answer
130 views

If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?

The random variables take values in $\mathbb{R}^d$. I have tried to prove this using characteristic functions. Let $\hat{\mu}_{X_n},\hat{\mu}_{Y_n},\hat{\mu}_{Z_n}$ be the characteristic functions of ...
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1answer
45 views

Why does the characteristic function always exist?

I've read that the characteristic function of a probability distribution always exists because it's bounded. However, the characteristic function is still Taylor expanded in terms of the moments of a ...
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0answers
22 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
19 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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1answer
36 views

How can you plot straight lines in Matlab using only values on x axis and the gradient of each line?

I'm trying to plot characteristics for the Burgers Equations. But I have to plot them only using the inbuilt function plot. It seems like a fairly straight forward problem but I still cant solve it
2
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1answer
52 views

Odd Inverse of a Characteristic function

I saw this formula today: $$ \mathbb{P} \left[ X > K \right] = \frac{1}{2} + \frac{1}{\pi} \int_0^\infty Re\left( \frac{e^{-iuK}\varphi(u)}{iu} \right) du $$ Where $\varphi(u)$ is the ...
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0answers
33 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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2answers
35 views

An integral related to Gaussian distribution

I am trying to evaluate the integral $$\int_{\mathbb{R}^d} e^{i \langle z,x \rangle} e^{- \langle x,A^{-1}x \rangle /2} dx $$ where $A$ is a positive-definite symmetric matrix. As a first step ...
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1answer
41 views

Inequality for Characteristic Function

For a discrete distribution the characteristic function $|\psi(u)|=1$ for other values of $u$ than $0$. We also know that $|\psi(u)|\leq 1$. How does this imply that for a continuous distribution we ...
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1answer
16 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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1answer
23 views

Solving a characteristic equation of a differential equation of the 4th order.

My DE equation looks like this: $y^{(3)}+y^{(2)}+34y^{(1)}+40y=xe^{-4x}+2e^{-3x}cos(x)$ I'm having trouble solving for the characteristic equation $r^4+r^3+34r^2+40r=0$ I got it down to ...
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1answer
46 views

Solve $h(x)+h'(x)(8-x)-32=0$ for x.

Solve $h(x)+h'(x)(8-x)-32=0$ for $x$.Where $$h(x)=\frac{\frac{1}{16}x^2 - 2 x + 80}{\left(\frac{1}{16}x^2 - 2 x + 20\right)^2}$$ Should I go with characteristic equations? or is there another way. ...
3
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1answer
52 views

Can anything be learned about a probability distribution *directly* from its characteristic function?

Some preliminaries: I know that one can take the inverse Fourier transform to get back the pdf...that is not what I am after. My question is whether the characteristic function, qua function, tells us ...
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1answer
38 views

Comparing sums of random variables

Consider $X_0,X_1\ldots,X_n$ mutually independent and $X_i \sim U(a_i,b_i)$. What is the probability that $\sum_{i=1}^n X_i<X_0$? Can you extend to mutually independent random variables with ...
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2answers
20 views

Convergence of a function involving a characteristic function of a decreasing interval

Let $f_k: \mathbb{R} \rightarrow \mathbb{R} , f_k(x) = \frac{1}{\sqrt{x}}\chi_{\left[\frac{1}{2^{k+1}},\frac{1}{2^k}\right]}(x)$ For $k \rightarrow \infty $ the interval ...
0
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2answers
46 views

Calculation of a characteristic function

Suppose $X_1, X_2, \ldots X_n \ldots$ are independent random variables with $$P(X_n = 1) = \frac{1}{2}$$$$P(X_n = -1) = \frac{1}{2}$$ Then $$\sqrt{\frac{3}{n^3}}\sum_{k=1}^n kX_k$$ converges to ...
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1answer
25 views

Is it possible to evaluate a normalizing constant for a characteristic function

Let $X$ be a random variable with density $f$ and characteristic function $\varphi$. Say we know $\varphi$ up to a constant $c$. Is it possible to evaluate this constant using $\int f(x)dx=1$ (or by ...
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0answers
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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1answer
47 views

Is a density function integrable?

Let $X \in \mathbb{R}$ be a continuous random variable with density function $f$ (i.e. $f(x)\geq 0$ and $\int f(x)dx=1$). Does this mean that $\int |f(x)|dx < \infty$ i.e. $f \in L^1$? (The reason ...
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0answers
54 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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1answer
35 views

Showing convergence by using Levy's continuity theorem

Let $X_n \sim \mathrm{Po} (n)$. I want to show that $Y_n = \displaystyle \frac{X_n -n}{ \sqrt{n}}$ for $n \rightarrow \infty$ converges in distribution to a random variable that is standard normal ...
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3answers
58 views

Proving that the characteristic function of $f(x)=\frac{1-\cos(x)}{\pi x^2}$ is $\max(1-|t|,0)$

How to prove that $\int_{\mathbb{R}}e^{itx}\frac{1-\cos(x)}{\pi x^2}dx=\max(1-|t|,0)$? Thank you!
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1answer
40 views

Characteristic function, how to integrate?

Find the characteristic function $\phi_X(t)$ of an absolute continious r.v. $X$ with density: $$f_X(x) = \frac{a}{2}e^{-a|x|} \qquad (a>0; x\in \mathbb{R})$$ Notation I have some ...
0
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1answer
40 views

What is the characterstic function?

Good evening, what is the characteristic function of a distribution with density function $$f(x) = \frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}},$$ while $x\in\mathbb{R}$ with parameter $\lambda > ...
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1answer
91 views

Find the characteristic function of combination of random variables

I have the same problem as here: Find characteristic function of random variable. Could you explain last equality? Can I get it without using law of total expectation? Update I have some idea, $E $$ ...
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1answer
45 views

Characteristics subgroups, normal subgroups and the commutator.

Let $G$ be a group and let $N$ be a normal subgroup of $G.$ Let $N'$ denote the commutator of $N.$ Prove that $N'$ is a normal subgroup of $G.$ What I do know is that the commutator subgroup is ...
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1answer
29 views

If the characteristics function of a random variable is differentiable even times then it has finite moment of even order

If the characteristics function of a random variable is differentiable $2n$ times then it has finite moment up to even order $2n$. We know the converse is correct, but how can we prove this statement? ...
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1answer
73 views

Weak Convergence If and Only If (Pointwise) Convergence of Characteristic Function

This is actually a theorem from lecture notes, with the corresponding proof. Unfortunately, it doesn't prove the last bit, or mention it at all (!), and I have a question about the penultimate bit. ...
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1answer
46 views

Carrying out the passage to the limit under an integral sign

For a sequence of distribution functions $(F_n)$ and their characteristic functions $(\varphi_n)$ I got $$ ...
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1answer
61 views

Characteristic function proof

How to prove that: $$ \frac{\exp(-x^2)+1}{2}$$ is a characteristic function? I will be grateful for any help!