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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31 views

Characteristic function of symmetric distribution

I know that if $X$ is a symmetric random variable, then the characteristic function is real, since $\overline{E[e^{itX}]} = E[e^{-itX}]=E[e^{itX}]$. However, is there a simple way (without appealing ...
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16 views

Proving two representations of a simple function are equivalent

Let's allow $\phi(x)=\sum^{n}_{j=1}a_{j}\chi_{A_{j}}(X)$ and also lets let $\phi(x)=\sum^{m}_{k=1}b_{k}\chi_{B_{k}}(X)$ be two different representations of the same simple function $\phi(x)$. Prove ...
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1answer
84 views

Show that if f (x) and f ′ (x) are not coprime, f (x) must have a multiple factor

Suppose $f(x)$ is in $F[x]$, $F$ a field, and the characteristic of $F$ does not divide the degree of $f (x)$. Show that if $f (x)$ and $f'(x)$ are not coprime, $f (x)$ must have a multiple factor ...
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2answers
110 views

Square of the Absolute Value of a Characteristic Function is a Characteristic Function

I am working on the following problem: Suppose that $\varphi(t)$ is the characteristic function for some random variable. Show that $|\varphi(t)|^2$ is a characteristic function. For what ...
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1answer
20 views

How to calculate characteristic function for this density: [closed]

$f(x) = \frac{1}{\pi}\frac{1 - {\cos}x}{x^2}$. I tried integrating by parts, by gotten nowhere. EDIT: Damn, that was easy. Thanks!
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36 views

Uniform Distribution Characteristic Function

What does it say about the uniform distribution that when generating the $n^{th}$ moments by the characteristic function we don't end up with cancellation of the imaginary values in the denominator ...
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0answers
22 views

Showing the sum of random variables are Linnik-Distributed

Suppose $X_{1},X_{2}$... are independent, identically Linnik(${\alpha}$)-distributed random variables and that N$\epsilon Fs(p)$ (First Success) and that N and $X_{1},X_{2}$...are independent. ...
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1answer
19 views

Is this a measurable function is this space?

Let $\Omega \subset R^n$ ($n\geq 2$) a bounded domain. Let $u \in L^{1}(\Omega)$. Let $F(x,k):= \chi_{\{ u> k\}}(x), (x,k) \in \Omega \times (-\infty , + \infty) $ . This function is measurable and ...
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1answer
36 views

Can I always decompose a random variable in sum of iid random variables?

Let $Z$ be a random variable. Can I always find a number $n \in \mathbb N > 1$, weights $w_i \neq 0$, and iid random variables $X_i$ such that $$Z = w_1X_1 + \dots + w_n X_n$$? Conversely, if I ...
3
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0answers
32 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 ...
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1answer
46 views

If $X_n$ sequence of random variables , equally distributed $EX_n=a, n=1,2,3…$ then $\frac{1}{n}\sum_{k=1}^{n}X_k\to^{P}a$

If $X_n$ sequence of random variables , equally distributed $EX_n=a, n=1,2,3...$ then $\frac{1}{n}\sum_{k=1}^{n}X_k\to^{P}a$ (convergence in probability) Proof: (using the fact that convergence in ...
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1answer
50 views

Random variable from characteristic function

I need to find the random variable knowing its characteristic function. I've found various post related to the problem, but the matter is that I cannot manipulate complex integral, neither I've ...
2
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1answer
54 views

Fubini-Tonelli and Lebesgue Integral exercise

Let $A,B \subset \mathbb R$ measurable subsets and $h:\mathbb R \to\overline{\mathbb R}$ defined s $h(x)=m((A-x) \cap B)$. Show that $h$ is measurable and that it satisfies $\int_{\mathbb ...
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0answers
31 views

Inverse Fourier formula for lattice random variables

Suppose that $X$ has a lattice distribution such that no strict sublattice of $\mathbb{Z}^d$ contains $X$ with probability $1$. Define the characteristic function of $X$ as usual: $$ \gamma(t)\equiv ...
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1answer
84 views

Find characteristic function of variable given a conditional distribution

Given $$ X \in Erlang(n,1)$$ and $$ Y|X=x \in Po(x) $$ How do I find the characteristic function of Y? I did start with this: $$ \phi_{Y|X=x}(t) = E[e^{itY} | X=x] = e^{x(e^{it}-1)} $$by the ...
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1answer
33 views

Distribution of the product of Cauchy IID random variables

Can anyone tell me if the product distribution of (say $n$) IID Cauchy random variables has a tractable form? And if so, what's a good way to go about deriving it? Characteristic functions (ie. ...
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2answers
103 views

Cantor Set : Characteristic function

What is the characteristic function of the C Cantor set? I already found that the function is Integrable but i couldn't find the exact result of the charateristic function of the C Cantor set.
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63 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 ...
0
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1answer
24 views

How to compute $E[e^{(i-1)Z}]$ where $Z$ is standard normal

Is there a trick on how to compute $$E[e^{(i-1)Z}]$$ where $Z$ is standard normal and $i$ is the imaginary number. We know that $E[e^{iZ}]$ is just a characteristic function of $Z$ but this is a ...
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2answers
20 views

$\lim 1_{A_n} = 1_{\cup A_n}$, where $(A_n)$ is increasing

Let $X$ be a set and $(A_n)$ be a sequence of increasing subsets of $X$. Show that: $$\large \lim_{n \to \infty} 1_{A_n} = 1_{\bigcup_{n=1}^{\infty} A_n}$$ Where $1_M$ is the characteristic function ...
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1answer
48 views

Confused about integral of characteristic function

It's usual to integrate functions like $\int_\mathbb{R} \chi_E(x)\ dm(x)$, where $E$ is a Borel set and $m$ is the Lebesgue measure on $\mathbb{R}$. In this case we have $\int_\mathbb{R} \chi_E(x)\ ...
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1answer
38 views

Does existence of a PDF imply the integrability of the characteristic function?

If the characteristic function of a random variable is integrable, then the random variable has a probability density function. Can anything be said about the other direction? That is, if a random ...
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1answer
37 views

Deducing expectation from characteristic function without using Lévy's formula

Consider a random variable $ X:[a,b]\rightarrow \mathbb R^+ $. We do not know the distribution of $ X $ but we know its characteristic function $ \phi_X(\ ) $. I know that with Lévy's formula I can ...
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1answer
23 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 ...
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1answer
18 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 ...
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2answers
83 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
68 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
33 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 ...
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1answer
36 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 ...
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0answers
39 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 ...
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1answer
35 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 ...
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1answer
12 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
64 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 ...
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1answer
34 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 ...
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1answer
65 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 $$ ...
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0answers
29 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 ...
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1answer
30 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
110 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 $\mathbb{R}^n$, ...
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1answer
42 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 ...
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2answers
160 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
60 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
85 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
107 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
103 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. $\phi_{\xi}(u) \neq 0\: \: \forall \: \: u \in \mathbb{R}$ ...
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1answer
62 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
56 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
43 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
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1answer
88 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
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1answer
55 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
46 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 ...