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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1answer
17 views

Computing the characteristic polynomial

Consider the following matrix A over the field $F_7$ $$ \left(\begin{array}{rrr} 3 & 4 & 4 \\ 2 & 5 & 2 \\ 1 & 2 & 5 \end{array}\right) . $$ I'm asked to ...
0
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1answer
17 views

$f_n(x) = \left\lfloor \frac{\sin(2\pi (x / n + 1/ 4) + 1 }{2}\right\rfloor$ and related

$f_n(x) = \left\lfloor \frac{ \sin(2\pi (\frac{x}{n} + \frac{1}{4})) + 1}{2}\right \rfloor = 1 \iff x = kn$ and $ f_n(x) = 0 \iff x \neq kn$. Let $g_n(x)$ be what's within the floor brackets. Then ...
3
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2answers
45 views

Characteristic function with modulus 1 implies degenerate distribution

Let $X$ be a random variable with characteristic function $\phi(\ )$ satisfying $|\phi(t)|=1$ for all $|t|\leq 1/T$ with some $T>0$. Show that $X$ is degenerate, i.e., there is $c$ such that ...
4
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0answers
41 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 ...
3
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1answer
26 views

Prove $f(x) = \int_{\Bbb {R}} \chi_E (y) \chi_E (y-x)dy$ is continuous, where $E$ is a subset of $\Bbb R$ with finite Lebesgue measure.

Prove $f(x) = \int_{\Bbb {R}} \chi_E (y) \chi_E (y-x)dy$ is continuous, where $E$ is a subset of $\Bbb R$ with finite Lebesgue measure. This looks like the convolution equation, so I consider $f$ as ...
0
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1answer
29 views

Fubini for Principal Value integrals in probability

Take a random variable $X$ with distribution function $F(x)=\mathbb{P}[X\leq x]$ and characteristic function $\phi_X(t)$. Then one can write ...
1
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0answers
32 views

Where to find “non-standard” characteristic functions?

Well, the title says it all. I need the characteristic function of the (generalized) arcsine distribution. I desperately searched the internet for it but haven't found anything. Is there some standard ...
1
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1answer
25 views

Characteristic function of a Poisson law

I would like to show that $$ \bigg(e^{itc} - 1 - \frac{it c}{1+ c^2} \bigg) \frac{1+ c^2}{c^2} G(c) - G(c-\delta)$$ is the characteristic function of a poisson distribution. This follows from the ...
0
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1answer
11 views

Convergence of the real part of the integral associated with a characteristic function

Let $I_n (t) = \int (e^{itu} -1)\frac{1+ u^2}{u^2}\, dG_n(u)$ be such that $$ I_n(t) \to \log f(t) $$ where $f(t)$ is the characteristic function of an infinitely divisible law. Why is it that $$ ...
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0answers
29 views

Characteristic function of $a/\sqrt(X+b)$ given characteristic function of $X$

Given that one knows the characteristic function of a rv $X$ how can we write the characteristic function of a function of $X$ when that function is $$\frac{1}{\sqrt{X+b}},$$ for $b$ constant? So, if ...
1
vote
1answer
31 views

What topologies are placed on the domain and range of the characteristic function?

under consideration is: $\mathbb{1}_{[0,1)}:\mathbb{R}\to\{0,1\}$ $$\mathbb{1}_{[0,1)}(x)= \begin{cases} 1,& 0\leq x<1\\ 0,& \text{otherwise} \end{cases}$$ My first question is that I don't ...
1
vote
1answer
41 views

$(X_n)_{n\in\mathbb{N}}$ independent Cauchy-distributed random variables. Convergence of $n^{-\gamma}(X_1+\cdots+X_n)$

I want to solve the following exercise but i am unsure if my ideas are correct or not. Let $(X_n)_{n\in\mathbb{N}}$ be i.i.d. random variables with probability density $$ ...
2
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0answers
24 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 ...
0
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1answer
22 views

Characteristic function of a non-negative random variable?

Is it possible to decide if a random variable is non-negative almost surely, by looking at the characteristic function of the random variable?
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2answers
39 views

Prove that $X,Y$ are independent iff the characteristic function of $(X,Y)$ equals the product of the characteristic functions of $X$ and $Y$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $X$ and $Y$ be random variables on $(\Omega,\mathcal A,\operatorname P)$ with values in $\mathbb{R}^m$ and $\mathbb{R}^n$, ...
0
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1answer
30 views

Characteristic Function and Convergence in Distribution of Sequence of R.V.

I am trying to solve the following: Let $X_1,X_2,...$ be a sequence of random variables with $P(X_n=\frac{k}{n})=\frac{1}{n}, k=0,1,2,...,n$. Find the characteristic function of $X_n$ and show that ...
1
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2answers
42 views

Find the elementary divisors of a matrix given its characteristic and minimal polynomials

This question comes from and old exam: Suppose the square rational matrix $A$ has characteristic and minimum polynomials $p_A(x) = x^6(x^2-2)^3(x^2+4)^2$ and $m_A(x) = x^2(x^2-2)(x^2+4)^2$ and $null A ...
3
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1answer
42 views

Computing Conditional Characteristic Function

I am trying to compute the characteristic function of the following: Let $X$ and $Y$ be random variables such that $Y\mid X = x\sim N(0, x)$ with $X\sim\mathrm{Po}(\lambda)$. Find the characteristic ...
3
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1answer
59 views

Show that $ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $ as $ \lambda \rightarrow \infty $

Given that the characteristic function for Y is $$ \varphi_Y (t) = e^{\lambda (e^{-t^2/2}-1)} $$ Show that $$ Y/\sqrt{\lambda } \xrightarrow{d} N(0,1) $$ as $$ \lambda \rightarrow \infty $$ I've ...
4
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1answer
105 views

Show $\int_{-\infty}^{\infty}\,f(u,t)dG(u)$ is a ch.f. where $G$ is a d.f. ; $f(u,\cdot)$ is a ch.f. and $f(\cdot,t)$ is continuous.

Show $$\int_{-\infty}^{\infty}\,f(u,t)dG(u)$$ is a ch.f. where $G$ is a d.f. ; and $f(u,\cdot)$ is a ch.f. for each $u$ and $f(\cdot,t)$is continuous for each $t$. Note that ch.f. means ...
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1answer
48 views

Characteristic function of an infinitely divisible distribution [closed]

I need to prove that for random variable $\xi$ which comes from infinitely divisible distribution characteristic function has no zeros, i.e. $\phi_{\xi}(u) \neq 0\: \: \forall \: \: u \in \mathbb{R}$ ...
0
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1answer
31 views

Recover the distribution of a Binomial random variable from its Characteristic Function

Hoping someone could show how to use the Characteristic Function of a binomial r.v. to recover its distribution. Using the inversion formula to recover the pdf of a r.v. with a continuous ...
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0answers
46 views

levy process and its characteristic function

Let $X(t)$ denote Levy Process. It can be proves that c.f of $X(t)$ is given: $E(e^{i\omega X(t)}) = e^{-\Phi(\omega)}$, where $ \Phi(\omega) = i \omega a - \int\limits^{-\infty}_{\infty} ...
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0answers
15 views

How to prove that the module of characteristic function is less than one

I would like to know if my resolution is right... I want prove than $|\varphi(t)| = \mathbb{E}[e^{i t X}]\leq 1$ , $\forall t \in \mathbb{R}$. $\it{proof:}$ First, note that $|\mathbb{E}[e^{i t ...
3
votes
1answer
70 views

Solving $u_t+u^2u_x=0$

I'm trying to solve the initial value problem with characteristis.: $$ u_t+u^2\cdot u_x=0\quad,\quad u(0,x)=f(x) $$ Where $u$ is a neat function with suitable requirements on its domain and its ...
0
votes
1answer
50 views

Why do characteristic functions use $e^{ix}$ and not $e^{-ix}$? Does it matter?

I've heard the characteristic function be described as the Fourier-Stieltjes Transform of the distribution measure of a r.v., but I was curious as to why it's written as $E[e^{ix}]$ and not the ...
2
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0answers
38 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 ...
1
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1answer
25 views

probability density and distribution functions

I have $6$ independent and identically distributed variables such that $C_i \sim N(1000,400)$. 1) Calculate the density functions, distribution function and characteristic function of $C = ...
2
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1answer
42 views

Equation with mean of random variables

In a proof I found the following conversion $$E\left[|X|\mathbf{1}_{[a,b]}(Y)\right] = E\left[|X|P(a \le Y \le b)\right]$$ I understand, why $E\left[\mathbf{1}_{[a,b]}(Y)\right] = P(a \le Y \le b)$, ...
1
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1answer
25 views

Using characteristic functions to establish convergence

So I have found the Characteristic function of the variable $X_ \lambda$ to be: $$\psi_{X_\lambda}(t) = \psi_{b(\lambda)(Y_\lambda-\lambda)}(t)=\mathbf Ee^{itb(\lambda)(Y_\lambda-\lambda)}=\mathbf ...
0
votes
1answer
39 views

Cauchy iid random variables and Strong Law of Large Numbers (helping understand)

Question: Let $(X_n)$ be a sequence of i.i.d. Cauchy random variables with density $\frac{1}{ π(1+x^2)}$. Use the characteristic function $φ(t) = e^{−|t|}$ of the Cauchy distribution to find the ...
2
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0answers
54 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?
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1answer
53 views

A reference for multi-dimensional characteristic functions

I'm looking for a well-written, rigorous and self-contained treatment of multidimensional characteristic functions, specifically Lévy's continuity theorem and the uniqueness theorem (which states that ...
1
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1answer
22 views

Differentiating Spitzer's identity

Let $(S_n)$ be an arbitrary random walk. Define $$M_n:= \max(0,S_1,...,S_n)$$ and $$S_n^{+} := \max\{0,S_n\}.$$ Spitzer's identity states that for $0<r<1$, we have $$\sum_{n=0}^{\infty} r^n ...
2
votes
2answers
41 views

Sequence of measurable functions $f_n=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$, uniform convergence

For each $n \in \mathbb N$, let $f_n:[0,\infty) \to \mathbb R: f_n(x)=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$. Show that there is no $E \subset [0,\infty)$ such that $|E|=0$ and $(f_n)_{n \geq 1}$ ...
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0answers
26 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
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0answers
61 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
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0answers
37 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, ...
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0answers
46 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 ...
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2answers
47 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 ...
0
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1answer
28 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
41 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
votes
1answer
91 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?
1
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1answer
32 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 ...
0
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1answer
72 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 ...
0
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1answer
17 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
50 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
42 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
votes
1answer
23 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 ...
1
vote
0answers
39 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 ...