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
22 views

Calculate characteristic function

$p(n)=(1-r)^2nr^{n-1},n=1,2,...$ $f(z)=1/(1-z)$ has derivative $f'(z)$ with convergent power series $f'(z)=1/(1-z)^2=1+2z+3z^2+...$ the answer I have got is $(1-r)^2e^{it}(1-re^{it})^{-2}$ , I am not ...
0
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1answer
23 views

Calculate the characteristic function $\varphi_W$ of W

$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise. I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.
1
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1answer
26 views

If $A$ is unitary and $f_A(x)=f_B(x)$ and $m_A(x)=m_B(x)$ then $A$ is similar to $B$

Given $A_{n\times n},B_{n\times n} \in \mathbb C$ then: if $A$ is unitary and the characteristic polynomial $f_A(x)=f_B(x)$ then $B$ is also unitary. if $A$ is normal and $f_A(x)=f_B(x)$ ...
0
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1answer
34 views

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar? So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is ...
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2answers
34 views

What probability mass, density or distribution function might corresponds to this moment generating function? [duplicate]

I have somehow come up with a random variable $X$ with moment generating function (assuming it exists) $$M_{X}(t) = -t (1 - e^t)$$ What is the probability mass, density or distribution function? It ...
1
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0answers
33 views

Solve IVP with the method of characteristics (quasilinear PDE) (+shockwaves)

The question i just can't figure out one bit is: Solve for the continious function $u(x,t)$: $$u_t+(1-2u)u_x=0 $$ $$-\infty < x < \infty, t>0$$ $$u(x,0)= \left\{ \begin{matrix} \frac{1}{4} ...
0
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1answer
19 views

Probability with Indicators Textbooks

I am new to using indicator functions (although I am quite familiar with undergrad-level probability and what an indicator function is). I am trying to relearn probability using indicator functions ...
3
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1answer
53 views

The partial derivative of a characteristic function (exercise).

Assume that you have a probability space $(\Omega, \mathcal{F},P)$ and a random varaible $X: (\Omega, \mathcal{F})\rightarrow(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$. Define the characteristic ...
6
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0answers
50 views

A random variable is symmetric if and only if its characteristic function is real-valued

Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric. Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega ...
0
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0answers
25 views

what are some drawbacks of the characteristic method for solving ODE?

for those who are unfamiliar with this method: http://imgur.com/a/kefvI What are some differential equations that can't be solved using this method? What are eventual disadvantages using methods. I ...
0
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0answers
19 views

$p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
1
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1answer
80 views

Characteristic function and moment generating function: differentiating under the integral

In order to justify the interchange of the derivative and integral when differentiating a characteristic function, one can use the dominated convergence theorem: $$\frac{d}{dt} \int e^{itx} P(dx) ...
2
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0answers
47 views

Proofs of some characteristic function properties

I saw these properties in the wikipedia page but I was unable to prove them. I had an idea to construct a random variable with the desired characteristic function, yet I haven't managed to that so ...
2
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1answer
62 views

Does $\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)$ imply independence of $X$ and $Y$? [duplicate]

It shouldn't, but I am blanking on a counterexample. ETA: Note that the $t$ is shared on both sides - which differentiates this from this question. Similarly $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ implies ...
2
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0answers
29 views

Characteristic function of $\chi^2$ distribution with $n$ degrees of freedom

I'm computing the formula for the characteristic function of the random variable $X \sim \chi^2(n), $ $n\in\mathbb{N}$. After some substitutions in the integral and some messing around with certain ...
0
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1answer
27 views

Calculating characteristic function of random variable

I would like to calculate the characteristic function of $Z_{\beta, n}=(1-\beta^2)^{1/2}\sum_{k=0}^n\beta^kX_k$, where $X_i$ are independent random variables with $P(X_i = 1)=P(X_i=-1) = 1/2$ and ...
0
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1answer
39 views

equivalence of properties of characteristic function of a random variable

I would like to prove that for a random variable $X$ and its characteristic function $\phi_X$ the following three properties are equivalent. $i) \ \phi_X(s) = 1$ for some $s \neq 0$ $ii) \ \phi_X$ ...
2
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1answer
31 views

Why $(1 - a|x|) \cdot I(|x| \leqslant \frac{1}{a})$ is a characteristic function?

I'm trying to prove Pòlya's theorem, but got stuck at the very first step (I was given a plan of the proof) This step is to proof, that $\phi_a(x) = (1 - a|x|) \cdot I(|x| \leqslant \frac{1}{a})$ is ...
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1answer
22 views

Two matrices $A, B$ satisfying in characteristic polynomial of $B$ and $A$, respectively.

Let matrices $A,B\in M_n(\mathbb{C})$ such that $A$ satisfy in characteristic polynomial of $B$, and $B$ satisfy in characteristic polynomial of $A$. Can we say that: $A$ is diagonalizable if and ...
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1answer
39 views

Characteristic function notation

Having a rather basic understanding of probabilities I would like to ask you what exactly means the following notation. I am looking at the Gardiner's handbook for stochastic methods and am interested ...
2
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2answers
40 views

Sam and Jane play with indicators.

There are $30$ red balls and $50$ white balls. Sam and Jane take turns drawing balls until they have drawn them all. Sam goes first. Let $N$ be the number of times Jane draws the same color ball as ...
3
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1answer
44 views

Why is a characteristic function continuous at $0$?

My lecture notes say: $t \mapsto \exp(-t^2/2)$ is a characteristic function (of $\mathcal{N}(0,1)$), so it is clear that it is continuous at $0$. So why does "being a characteristic function" ...
2
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1answer
44 views

Characteristic exponent of $\alpha$-stable Levy process

I'm studying the book "Probability and Stochastics" by Erhan Cinlar - the probability class that I took covered until the beginning of chapter 7, and I'm now trying to do some of the exercises but ...
2
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0answers
47 views

Distribution of $aX+bX^2+cX^3$ where $X$ is standard normal

I am looking for some distributional characteristic (for example a characteristic function) of a random variable which is defined as $aX+bX^2+cX^3$, where $X$ is a standard normal variable. Is there ...
5
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1answer
86 views

Tensor product for vector bundles is commutative, associative, and has an identity element?

How do I see that the tensor product operation for vector bundles over a fixed base space is commutative, associative, and has an identity element?
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0answers
42 views

Characteristic function of Cantor distribution

In Wiki, it provided the Characteristic function of Cantor distribution. That is, $e^{\mathrm{i}\,t/2}\prod_{i= 1}^{\infty} \cos{\left(\frac{t}{3^{i}} ...
7
votes
1answer
49 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
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1answer
22 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 ...
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1answer
44 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
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2answers
19 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
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1answer
62 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: ...
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0answers
45 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
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1answer
14 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
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1answer
38 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?
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0answers
53 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} ...
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2answers
36 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 ...
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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
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1answer
35 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 $1^3$, $(-2)^3$ , ...
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0answers
47 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
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1answer
40 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
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0answers
49 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 ...
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0answers
23 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
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1answer
42 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
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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
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1answer
40 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
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1answer
20 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
91 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
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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
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1answer
51 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
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1answer
16 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) ...