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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1answer
60 views

How to show the following characteristic function is positive definite

Let $\phi$ is the characteristic function of a probability measure on $\mathbb{R}$, how can I prove $\sum_{i,j=1}^{n}{\phi(t_i-t_j)\bar{\phi}(t_i-t_j)\xi_i\bar{\xi_j}}\geq 0$ for all $n\in\mathbb{N}$, ...
3
votes
1answer
26 views

difference of characteristic function for measure and random variable

Suppose random variable $X$ follow a certain (known) distribution. And I denote the probability measure $\mu$ as the distribution (pushforward measure) of $X$. Is there any difference between ...
1
vote
1answer
48 views

Convolution of indicator functions with values in a finite field

This is something I haven't seen online yet, indicator functions with values in a finite field. Probably for a good reason, but I would like to know why, and if there are still things that can be ...
0
votes
2answers
21 views

Characteristic and Principal Ideal.

This might be a simple question for some of you, but I am quite confused on the whole concept of principal ideals. Question 1: What is the characteristic of $\mathbb{Z}_2[X,Y]$ where it is the ring ...
4
votes
1answer
65 views

How to calculate $\int_0^1\cdots\int_0^1\frac{1}{x_1+x_2+\cdots+x_n+1}dx_1\cdots dx_n$

I met an integral $$\int_0^1\cdots\int_0^1\frac{1}{x_1+x_2+\cdots+x_n+1}dx_1\cdots dx_n$$ I calculated $n=1,2,3$ and made an induction! then i got the result: ...
1
vote
0answers
46 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} ...
0
votes
1answer
15 views

Expected value of number of “good” triples in the set of vectors

Given m vectors $v_1, v_2, ...,v_m$ in $\mathbb{R}^n$ with coordinates $0$ or $1$. Each vector is a result of $n$ Bernoulli trials. It may be that $v_i = v_j, i \neq j$. Consider three vectors: ...
1
vote
1answer
39 views

Characteristic function using conjugate property

To prove that $e^{−i|x|}$ is not a characteristic function: $$e^{−i|x|} =\cos|x|-i \sin|x|.$$ Its conjugate will be $\cos|x|+i \sin|x|$ which is not equal to $\phi(-x)$. Is my solution correct?
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} ...
1
vote
2answers
37 views

Aymptotic Convergence of Mean Estimator

I want to show: Let $x_{i}$ be an iid random variable with support $x_{i} \in [0,1]$. Prove $n^{1/3}\frac{1}{n} \sum\limits_{i=1}^{n} (x_{i} - \mathbb{E}[x_{i}] ) \xrightarrow{p} 0$. From ...
-1
votes
1answer
37 views

Let $M$ be an $n\times n$ diagonalizable matrix with characteristic polynomial, find the matrix

Let $M$ be an $n\times n$ diagonalizable matrix with characteristic polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$, Find the matrix $a_nM^n+a_{n-1}M^{n-1}+...+a_1M+a_oI$, where $I$ is the ...
1
vote
1answer
37 views

Find the characteristic polynomial of $(M^{-1})^3$

Given that M is a square matrix with characteristic polynomial $p_{m}(x) = -x^3 +6x^2+9x-14$ Find the characteristic polynomial of $(M^{-1})^3$ My attempt: x of $(M^{-1})^3$ is $1^3$, $(-2)^3$ , ...
0
votes
0answers
51 views

Calculation of infinite product

My question is to prove the identity: $$ \prod_{n=1}^{\infty}\left(\frac{\cos t-1}{n}+1\right)=\exp\left(-\int_0^1x^{-1}(1-\cos xt)dx\right) $$ which arises as a product of characteristic functions of ...
1
vote
1answer
47 views

Convolution of characteristic function

I am trying to figure out following problem. Let A ⊂ R. Then we can define the characteristic function: Let a be bigger than 0. I am trying to find a following convolution: \begin{align} ...
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 ...
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? ...
3
votes
1answer
44 views

Abstract machines that compute primitive recursive functions

What it the simplest (least powerful) abstract machine that can compute primitive recursive sets, i.e. sets whose characteristic or indicator function is primitive recursive? ...
3
votes
2answers
53 views

Show that if $a_n(X_n-X) \overset{\mathcal{D}}\to Z$ then $X_n \overset{P}\to X$.

Let $(a_n)\subseteq \Bbb{R}$ be a sequence such that $a_n \to \infty$. Let $(X_n)$ be a sequence of random variables such that $a_n(X_n-X) \overset{\mathcal{D}}\to Z$ fore some random variables $X$ ...
0
votes
1answer
43 views

Which random variable has the characteristic function $f(t)=\frac{e^{it}}{1-it}$

Which random variable has the characteristic function $$f(t)=\frac{e^{it}}{1-it}$$ This is quite important for me to know, I know I have seen it somewhere, but I cant remember which random variable.
0
votes
1answer
30 views

Derivatives of characteristic function

Let $\phi$ be the characteristic function for random variable $X$. I know that if $E [|X|] < \infty$, then dominated convergence implies existence of the first derivative, and in particular, ...
2
votes
1answer
100 views

Asymptotic standard normal distribution

I need to solve the following exercise. Assume that $X_\lambda$ is Poisson distributed with mean $\lambda$ . Show that $Y(\lambda) = \frac{X_\lambda - \lambda}{\sqrt{\lambda}}$ is asymptotic ...
2
votes
1answer
49 views

Convergence in probability using characteristic functions

I need to solve the following problem. Let $X_1,X_2,\dots$ be independent random variables all with expectation $0$ and variance bounded by $M$. Prove that $\frac{1}{n}\cdot \sum\limits_{k=1}^{n} ...
0
votes
1answer
63 views

Analytical continuation of moment generating function

Let's say some distribution $F(t)$ has finite moment generating function on an open ball (-R, R). $M(x) = \sum m_n x^n /n!$ Let's extend $M(x)$ to $M(z)$ on a complex strip $S = \{z| |Re(z)| ...
0
votes
1answer
20 views

Explanation of pointwise convergence for this particular characteristic (indicator) function

Let $f_{n}(x)=\chi_{[n,\infty)}(x) = \begin{cases} 1 & \text{if}\,x\in[n,\infty)\\ 0 & \text{if}\, x \notin [n,\infty) \end{cases}$. According to my textbook, as $n \to \infty$, $f_{n}(x) ...
4
votes
0answers
110 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 ...
1
vote
1answer
44 views

What is the function that solves the Cauchy Problem?

Solve the Cauchy Problem $u_x+(x+y) u_y=1, u(x,-x)=0$ using the method of characteristics. I arrived the $c_2=u-x$ and $c_1=e^{(-x)}(y+x+1)$. Then $c_2=G(c_1)$, and using the initial conditions I ...
1
vote
0answers
39 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 ...
2
votes
1answer
90 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 ...
1
vote
2answers
149 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 ...
1
vote
1answer
21 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!
2
votes
1answer
30 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. ...
1
vote
1answer
20 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 ...
4
votes
1answer
38 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 ...
4
votes
0answers
45 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 ...
1
vote
1answer
50 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 ...
1
vote
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
votes
1answer
61 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 ...
0
votes
0answers
38 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 ...
0
votes
1answer
128 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 ...
0
votes
1answer
53 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. ...
1
vote
2answers
144 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.
1
vote
0answers
64 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
votes
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 ...
0
votes
2answers
21 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 ...
1
vote
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)\ ...
1
vote
1answer
44 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 ...
2
votes
1answer
42 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 ...
0
votes
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 ...
0
votes
1answer
19 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
votes
2answers
107 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 ...