# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $\mathbf{t}\... 2answers 72 views ### Is the product of two different characteristic functions also a characteristic function? Suppose that$\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_{...
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]}...