Questions tagged [characteristic-functions]

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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11 views

Find the limit cdf for a sequence of random varibales whose characteristic functions do not converge.

X is a random variable with $P(x=-1)=P(x=1)=\frac{1}{2}$, $Y_n=nX$ is a sequence of random variables, n=1,2, $\dots$ $S_n = \frac{1}{n}\sum_{i=1}^{n} Y_i$ is the partial sum of $Y_n$. I am asked to ...
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$X \sim U[-1,1]$, is there $Y$ independent of $X$ s.t $X+Y$ and $\frac{Y}{2}$ have the same distribution?

I have thought that if there exist such $Y$ then we can look at characteristic functions of $X + Y$ and $\frac{Y}{2}$ to get: $$ \phi_{X+Y}(t)= \phi_X(t)\phi_Y(t) = \phi_{\frac{Y}{2}}(t) \\ \frac{\...
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Characteristic Function as a Fourier Transform

The fourier transform of a function is defined to be: $$\hat{f}(\omega)=\int_{R}e^{-it\omega}f(t)dt$$ which I understand that essentially $e^{-it\omega}$ controls the frequency at which our ...
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1answer
27 views

Uniform Continuity of Characteristic Function

I am trying to understand the concept of uniform continuity as it pertains to characteristic functions. First my understanding of uniform continuity: Def: $$\forall x_0, \forall \epsilon>0, ...
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1answer
18 views

Problem calculating the characteristic function of the exponential distribution

I was trying to calculate the characteristic function of the exponential distribution $$\varphi(t) = \mathbb E[e^{itX}] = \int_{-\infty}^\infty e^{itx} \lambda e^{-\lambda x} \cdot 1_{[0,\infty)}(x) \,...
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Showing a Random Variable is Discrete from the Characteristic Function

Question: The characteristic func. of a r.v. X is $$\dfrac{e^{it}(1-e^{nit})}{n(i-e^{it})}$$ Show that X is a discrete r.v with $p(x)=\dfrac1n$ for $x=1,2,\cdots n.$ Can one please help me to ...
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$X_p \sim GEO(p)$. What's the limiting distribution of $X_p$ as $p \to 0+$? (Should be $EXP(1)$)

I've been given the above question, and I've tried getting $EXP(1)$, but no matter what I try fails. We've learned about Characteristic Functions, so I assume we need to show that the $$\lim_{p \to ...
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1answer
34 views

Is this random walk transient or recurrent

Suppose $\mathbb{P}(X=1)=\mathbb{P}(X=-1)=\frac{1}{6}$ and $\mathbb{P}(X=0) = \frac{2}{3}$. Is the random walk of $X$ transient or recurrent? I wanted to use well known result that random walk is ...
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55 views

Conditions for obtaining the characteristic function from MGF

Notation used is taken from Gallager's text on stochastic processes. For a random variable $X$, let $g_X(r)=\mathbb{E}[\exp(rX)]$ be its moment generating function where $r$ is a real number. ...
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39 views

How can I find the characteristic function in this question?

This is a question from my university list: Let $\{x_1, x_2, \ldots, x_n\}$ be statistically independent and identically distributed random variables, each with an exponential probability density ...
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Differences Between Characteristic Function and Moment-Generating Function

Why can't the moment generating function be defined for all random variables, while the characteristic function can be defined for all random variables?
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Mode directly from a complex-value characteristic function

I have the following characteristic function: $$ \phi(t) = \phi_1(t)\phi_2(t). $$ I know characteristic functions $\phi_1(t)$ and $\phi_2(t)$ and the corresponding PDFs. Nevertheless, I cannot ...
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How can I determine the characteristic function of $Z = \sum_{j = 1}^{N}\frac{X_j}{10^j}$?

I am currently working on an exercise regarding characteristic functions. Consider a set of i.i.d. random variables, $\{X_1,\ldots,X_n\}$, uniformly distributed on $\{0,1,2,\ldots,9\}$. I want to ...
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40 views

Why do we use exponents in characteristic function

A student who is attending probability 101, learned about normal distribution and generating functions recently. We are given a "generating function" as follows: $$G(t)=<e^{itx}>=\int_{-\infty}^...
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Getting probability density function from a complex characteristic function

I have been trying to convert the characteristic function of the chi-squared distribution: $$\phi(t) = (1-2it)^{-k/2}$$ to its probability density function using the following equation: $$f(x) = \...
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Sums of Cauchy distributions and their conditional distribution

It's well-known that sums of independent Cauchy are Cauchy. Indeed, for $Y_1 \sim \text{Cauchy}(m_1, s_1)$ and $Y_2 \sim \text{Cauchy}(m_2, s_2)$, we have characteristic function \begin{align*} \...
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if every continuous characteristic function is constant then ,M is connected

prove for a metric space $M$, if every continuous characteristic function is constant then $M$ is connected. I actually know how to prove the other direction, but I do not know how to work on this ...
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Prove about sequence of sets and the characteristic function.

I'm trying to prove: If $(E_{n})_{n\geq 1}$ is a sqeuence of sets of $X$. We define $D_{1}=E_{1}$ and $D_{n}=D_{n-1}\bigtriangleup E_{n}$ for every $n \in \mathbb{N}$ then $\varliminf D_{n}=\...
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Approximation of Indicator Functions by Bounded Continuous Functions, in Euclidean Space.

Let $A\subset \mathbb{R}^n$ be open. Define the indicator function $$ I_A(x)=\begin{cases} 1 & \text{if}~x\in A\\ 0 & \text{otherwise}. \end{cases}$$ Does there exist a sequence $f_n : \...
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In the multivariate case, does an integrable characteristic function implies a continuous density function?

In the univariate case, it is easy to see that if the characteristic function $\phi$ of a $\mathbb R$-random variable $X$ is integrable, then by the inversion theorem the p.d.f $f$ of $X$ is ...
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27 views

Finding a characteristic function that is strategically equivalent

I consider the pollution game. There are five villages around a lake. Each village takes its drinking water from the lake and discharges its sewage into the lake. Each faces the choices of whether to ...
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If $A$ is an $n$-by-$n$ matrix with characteristic polynomial $(-1)^nt^n+b_{n-1}t^{n-1}+\cdots+b_1t+b_0$, show that $tr(A)=(-1)^{n-1}b_{n-1}$ [duplicate]

Let $A$ be an $nxn$ matrix with characteristic polynomial $$f(t)=(-1)^nt^n+b_{n-1}t^{n-1}+\cdots+b_1t+b_0$$ Show that $tr(A)=(-1)^{n-1}b_{n-1}$ I think I should be able to get this from the ...
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Prove that function is measurable

Prove that the following function is measurable: $$f = \sum_{n=1}^{\infty} n \chi_{[n,n+1]}$$ where $\chi$ is the characteristic function. I am aware of the definition of a measurable function, ...
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1answer
95 views

Estimating Fourier transform of an indicator function

Given a multiplicative subgroup $ \Gamma \subseteq F^*_p $ (multiplicative group of integers modulo prime $ p $), its indicator function $ \Gamma(x) $, and the Fourier transform of a function $ f: F_p ...
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33 views

Characteristic polynomial for A when $A=(a_{ij})$

Let $A$ be an $n \times n$ matrix with characteristic polynomial $f(t)=(-1)^nt^n+b_{n-1}t^{n-1}+\cdots+b_1t+b_0$ Define $A=(a_{ij})$. Show that $f(t)=(a_{11}-t)(a_{22}-t)\cdots(a_{nn}-t)+q(t)$, where ...
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Please verify my proof for the lemma: “If $X_n \to_d X$ and $X_n + Y_n \to_d X$ with $X_n$ and $Y_n$ independent for each $n$, then $Y_n \to_p 0$”

It is the lemma 5.1 in this paper: https://dornsife.usc.edu/assets/sites/1193/docs/lin.pdf, and the paper contains the proof. My attempt is: By Portmanteau theorem, $X_n \to_d X$ is the equivalent ...
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Need help understanding a proof on sets of divergence

I have few questions regarding the proof to the theorem : ( Katznelson "An Introduction to Harmonic Analysis" Chapter 2.3 What I am struggeling to understand is the last bit of the proof: Why does $ ...
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71 views

Prove that this set is a ring

Let $X$ be a set and $\mathcal{B}$ a subset of the set of maps from $X$ to $\mathbb{Z}_2$. For any subset $A\subset X$ we define the characteristic mapping of A as the mapping $\chi_A:X\...
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$A\in M_3(\mathbb R)$ with $A^8=I$, then what can we tell about the degree of its minimal polynomial?

$A\in M_3(\mathbb R)$ with $A^8=I$, then what can we tell about the degree of its minimal polynomial? What I understand that $A$ satisfies $x^8-1=0$, and as we know characteristic polynomial ...
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Showing that if X and Y are independent and have chf $\phi$, and distribution $\mu$, then a property holds.

Ex 3.3.2 is posted below: Ex 2.1.5 says that $P(X - Y = 0) = 0$, which means $P(X = Y) = 0$ for $X$ having continuous distribution. I am not sure how to proceed and do this problem. $|\phi(t)|^{2}$ = ...
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All derivatives of the holomorphic function $g(z)$ vanish at $z=0 \implies g\equiv 0$.

I apologice in advance if this question is too trivial. Let $\nu$ be the standard gaussian measure and $f\in L^2_{\nu}(\mathbb R)$. Let \begin{equation} g(z)=\int_{\mathbb R} e^{i x z} f(x)\nu(d x)=\...
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1answer
63 views

Convergence of a series a.e

In the following exercise, they supposed that $(X_n)_n$ is a sequence of independent and identically distributed random variables. If we supposed that the distribution is degenerate, then $\exists c ...
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45 views

How to find the subset of a characteristic function?

For example, given the characteristic function XA + XB - XA $ \cap $ B where A and B and subsets of set S. How to find the subset of the characteristic function?
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Finding the characteristic function of a 2nd order ODE with variables on both sides

The given function is $$x^2y^{''}+xy^{"}+y=\ln(x)$$ Now finding the characteristic or aux. function is very simple, usually. This problem has $x$ terms on both sides of the equation. Would it be ...
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1answer
37 views

Characteristic functions and metric spaces

Let $\mathcal{P}$ be the space of probability measure on $\mathbb{R}.$ Define $d(\varphi,\phi)=\sup_x|\varphi(x)-\phi(x)|/(1+|x|),$ where $\varphi$ and $\phi$ are the characteristic functions of two ...
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1answer
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Expanding to second order

I have the following characteristic function of S. I understood how I got to this characteristic function yet the next part I want to show that as $$ n \rightarrow \infty, \varphi_S (t) \rightarrow ...
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1answer
37 views

Characteristic Function of Sum and Difference of Non-Identical Exponential Random Variables

This question is related to an answer of this question. In the linked question, three independent (non-identical) exponential random variables $X,Y,Z$ with means $\mu_X,\mu_Y,\mu_Z>0$ are ...
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Degenerate two-dimensional distribution

Assume $\phi$ is one-dimensional characteristic function, $a_1$, $a_2$ are arbitrary constants. How can we show that $\phi(a_1\xi_1 + a_2\xi_2)$ as function of two variables $\xi_1$ and $\xi_2$ is two-...
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1answer
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Characteristic function of multiplication of two independent r.v.s

$X$ is r.v. with Exponential with $\lambda=1$ and Y has a characteristic function $e^{-|t|^a}$ for $a\in (0,2)$. What is the characteristic function of $Z=X^{\frac{1}{a}}Y$ if $X$ and $Y$ are ...
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1answer
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Show that $E|X|^p$ is finite

Show $\varphi(t) = \frac{1}{1 + |t|^{\beta}}$ is a characteristic function for $\beta \in (0,1]$. Let $X$ be r.v. with characteristic function $\varphi$. Prove that $E|X|^p < \infty$ for all $0 <...
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1answer
56 views

How can I prove that $\varphi(t)=e^{-|t|^{\alpha}}$ is not a characteristic function for $\alpha > 2$

I am trying to prove that for $\alpha > 2$ this is not a characteristic function $$ \varphi(t)=e^{-|t|^{\alpha}} $$ Suppose $t > 0$. Differentiating twice gives us $$ (e^{-t^{\alpha}})^{(2)} = \...
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1answer
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Boundedness of characteristic function (using Fourier transform)

Let $\mu$ be a probability measure on $\mathbb{R}^n$. Its Fourier transform is denoted $\mu'$ is a function on $\mathbb{R}^n$ is given by $$ \mu'(u)= \int e^{i\langle u,x\rangle} \mu(dx) $$ Now it ...
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53 views

if $\phi$ is a characteristic function, then $|\phi|^2$ is also a characteristic function [closed]

Show that if $\phi$ is a characteristic function of some random variable, then $|\phi|^2$ is also a characteristic function.
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$\sigma$-Algebra generated by a simple characteristic function

Suppose that $X:\Omega \rightarrow \mathbb{R}$. Let $A \in \mathcal{B}$ be a Borel set. Suppose that $X = \sum_i a_i\chi_{A_i}$ is a simple random variable defined on the probability space $(\Omega,\...
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Indicator function and fractional Sobolev spaces

I am wondering, and I don't have an exact answer - what is the optimal fraction that the characteristic function $\chi_\Omega$ does not lie in $H^s (\mathbb{R}^d)$ for any $\Omega$ of positive measure?...
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prove characteristic function of an open set is continuous.

The question is asking : prove $\chi_G$ is continuous at each point of $G$, where $G$ is an open set in $R$. it may seem silly! but I ended up proving $\chi_G$ is NOT continuous. $\chi^{-1}(0)=G^c$ $\...
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1answer
48 views

If $A^3+2A^2+2A+I_n=0_n$, then $\det(A)=-1$, where $A$ is a square matrix of odd size, with real entries

If $A^3+2A^2+2A+I_n=0_n$, then $\det(A)=-1$, where $A$ is a square matrix of odd size, with real entries, and $0_n$ is the zero matrix. Let us denote the identity matrix by $I_n$. What I have ...
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1answer
25 views

Product of two characteristic functions

Suppose that $\mathbf{X}$ is a $\mathbb{R}^k$-valued random vector and $\mathbf{Y}$ is a $\mathbb{R}^\ell$-valued random vector. $\mathbf{X}$ and $\mathbf{Y}$ are not independent. For any $\mathbf{t}\...
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2answers
72 views

Is the product of two different characteristic functions also a characteristic function?

Suppose that $\phi_{X}(t)$ and $\phi_{Y}(t)$ are characteristic functions of $X, Y$, respectively. Moreover, $X$ and $Y$ are NOT independent random variables. I want to know if $\phi_{X}(t)\cdot\phi_{...
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34 views

Double Integrals of Characteristic Functions

I was working on the double integral of a characteristic function and was wondering if I had properly set it up. I want to integrate $$\int_0^{b_2}\int_0^{s_2}(s_2-s_1)\chi_{[0,b_1]}(s_1)\chi_{[0,b_2]}...

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