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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How to compute the characteristic function

Let $X_n$ be a sequence of identically distributed, independent random variables and let the distribution of $X_n$ be $\mu_{X_n}(\{1\}) = a$ and $\mu_{X_n}(\{-1\}) = 1-a$ for 0 < a < 1. ...
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1answer
31 views

Fatou Lemma: Why is $\lim\inf f_n = 0$ where $f_n = \chi_{[n,n+1]}$

In this wildly popular post, there is a claim: I like to remember this by example; specifically let $f_n = \chi_{[n,n+1]}$. Then $\lim \inf f_n = 0$, and $\lim \inf \int f_n = 1$. So $f_n = ...
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30 views

Stability and characteristic roots of difference equations

I often hear "For a process to be stable its characteristic roots or poles must be outside the unit circle (for casual process)". All right, consider the recurrence relation: ...
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1answer
21 views

What is the eigenvalue for characteristic polynomial $(1+\lambda^2)\lambda$?

What is the eigenvalue for characteristic polynomial $(1+\lambda^2)\lambda$? Is it $0,i,-i$ or $0,i,i$?
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29 views

How to prove convergence for the following?

Let $X_m$ be distributed as follows: $$\mathbb P(X_m=\pm m)=\frac{1}{2m^\beta}, \mathbb P(X_m=0)=\frac{m^\beta-1}{m^\beta}.$$ Let be $S_n=\sum_{m=1}^{n}X_m$, then I. If $\beta>1$, then ...
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2answers
26 views

How would I find the characteristic equation of this Recurrence Relation?

Find and solve a recurrence relation for the number of $n$-digit ternary sequences with no consecutive digits being equal. Since for ternary, meaning only $3$ possible entries for each space, e.g. ...
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1answer
445 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
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1answer
23 views

How can I make sure that the classical way of calculating the characteristic function of an exponential holds?

Given $f(x)=\lambda e^{-\lambda x}$, I want to find $\phi(t) = E(e^{itx})$ (characteristic function). Classical way: \begin{align} \phi(t) &= \int_0^{\infty} e^{itx}\lambda e^{-\lambda x} dx \\ ...
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1answer
14 views

How to reduce into canonical form

Determine the type of the following equation and reduce the PDE to its canonical form $u_{xx} + 4u_{xy} + 4u_{yy} + u = 0$. We consider pdes in the form ...
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1answer
25 views

Finding a characteristic function of an exponential pdf

My pdf is defined as follows: $$f_X(x) = \frac{1}{\tau} e^{-x/\tau}$$ At first I started finding the characteristic function like so: $$\hat{f}_X(\xi) = \mathbb{E}[e^{i\xi X}] = ...
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27 views

Difference between the characteristic function of a sum of independent vectors and a Gaussian

In the book "Sums of Independent Random Variables" by Petrov the following lemma appears in page 109, in preparation to prove the Berry-Esseen inequality Let $X_1,...,X_n$ be independent random ...
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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. ...
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1answer
34 views

Characteristic functions of infinite dimensional random elements

I am trying to understand if it is possible to prove the convergence in distribution of a sequence of infinite dimensional random elements using characteristic functions. Suppose that ...
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1answer
2k views

How to get PDF from characteristic function

I would appreciate if anybody could explain to me with a simple example how to find PDF of a random variable from its characteristic function. Thank you.
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1answer
48 views

Analogous result for basic Central Limit Theorem

Let $X_1, X_2, ..., X_k$ be an independent and identically distributed random variables. Assume $\mathbb{E}(X_i^2) < \infty$ for $1 \leq i \leq k$ and $$\frac{X_1 + X_2 + ... + X_k}{\sqrt{k}} \ ...
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63 views

Properties of characteristic functions under statistical dependence

Given random variables $X,Y,Z$,and $\phi(.)$ denoting the characteristic function, I can see that the following is true when $Z$ is independent of $X,Y$: $|\phi_{X+Z,Y} (t, s) − \phi_{X+Z}(t)f_{Y} ...
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1answer
31 views

Given the characteristic equation of A.Find equation for B.

Given that $\chi_A(x)=x^3-ax^2+bx-c$ Find $\chi_B(x)$ For: a)B=A-2I b)$B=A^2$ For a) would you put x+2 in for x in the $\chi_A(x)$. As Det(XI-B)=Det(XI+2I-A)= det((X+2)I-A) And b: Im not sure ...
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52 views

A sum of a random number of Poisson random variables

in my probability class I was given this question on which I am stuck concerning a sum of random number of Poisson random variables: Let us define the countable set of independent random variables ...
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3answers
57 views

What is the integral of the function $f = \chi_{[0,\infty)}e^{-x}$

I have $$\int_{\mathbb R} f\,d\mu = \lim_{n\to\infty}\int_{[0,n]} f\,d\mu$$ So $$\begin{align}\int_{\mathbb R}f\,d\mu &= \lim_{n\to\infty}\int_{[0,n]}\chi_{[0,\infty)}e^{-x}\\ ...
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54 views

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

I'm looking for the answer to the problem in the title. I know it comes from the Marcinkiewicz theorem, but I need formal proof of it.
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27 views

Example for sum of dependent random variables of which the characteristic function can be factorized.

The characteristic function of two independent random variable can always be factorized, but the opposite is not true: I would like to construct an example, where the characteristic function can be ...
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1answer
63 views

Characteristic function

Let $\{X_n:n\ge 1\}$ be a sequence of i.i.d. Bernoulli random variables with probability of success $0<p<1$, i.e, $$P\{X_1=1\}=1-P\{X_1=0\}=p.$$ The random variable Y is independent of ...
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35 views

Are the following functions characteristic functions of a random variable?

Let be $X$ an arbitrary random variable with characteristic function $\varphi(t)$. Prove that $$\psi(t)=\frac{1}{2-\varphi(t)},$$ and $$\chi(t)=\int_{0}^{\infty}\varphi(st)e^{-s}ds$$ are also ...
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characteristic method for wave equation in a non-uniform string

wave equation in a non-uniform is $u_{tt}=c(x)u_{xx}$, $c(x)=1$, when $0<x<1/2$, $c(x)=2$, when $1/2<x<1$, Does $u(x,t)$ has the form like $u(x,t)=f(x-t)+g(x+t)$ when $0<x<1/2$, ...
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Finding the odometer function of an abelian sandpile

As a computer scientist and "armchair" mathematician, I'm trying to replicate the images found here of Abelian sandpiles on a square lattice, where the initial configuration is $n$ chips on a single ...
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205 views

Does “independence” of moments imply independence?

Suppse you have two random variables $X,Y$ and you are given that for any $m,n$ that: $$E(X^n Y^m) = E(X^n)E(Y^m)$$ Does this imply that $X$ and $Y$ are independent? Are there some condtions on how ...
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1answer
35 views

Characteristic functions, show that $\mathrm{E}[X_1 + X_2 +…+X_n]^3 = \mathrm{E}[X_1^3] +…+\mathrm{E}[X_n^3]$ when $\mathrm{E}[X] = 0$

I want to show that for independent variables X_1, X_2,...,X_n, with expectation 0 and finite third moments, $$\mathrm{E}[X_1 + X_2 +...+X_n]^3 = \mathrm{E}[X_1^3] + \mathrm{E}[X_2^3] ...
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Convergent sequence of characteristic functions with continuous integrable limit.

Suppose $\varphi_{n}$ is a sequence of characteristic functions converging pointwise to a function $\varphi$ which is continuous and absolutely integrable. I want to show that $\varphi$ is the ...
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1answer
30 views

Prove that $u^{-1}\int_{-u}^u(1-\varphi(t))dt\to 0$ as $u\to 0$

How to prove that $u^{-1}\int_{-u}^u(1-\varphi(t))dt\to 0$ as $u\to 0$ $\varphi(t)$ is the characteristic function of a random variable, if $u\to 0$ then $\varphi(t)\to 1$, so one gets $\frac00$ ...
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1answer
82 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}} ...
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1answer
23 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 ...
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1answer
25 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.
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1answer
27 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 $f_A(x)=f_B(x)$ ...
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1answer
36 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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1answer
90 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) ...
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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} ...
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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? ...
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1answer
23 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 ...
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54 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 ...
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1answer
60 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 ...
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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 ...
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34 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 ...
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$p$-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 ...
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1answer
41 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 $s \neq 0$ $ii) \ \phi_X$ ...
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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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76 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 ...
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33 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 ...
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1answer
34 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
30 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 ...
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1answer
24 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 ...