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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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Characteristic function of exponential distributed random variable

Given: $$f_X(x) = \lambda e^{-\lambda x},\; x\in X$$ Wanted: The corresponding characteristic function $\phi(ju)$. \begin{align} \phi(ju)&=\mathbb{E}(e^{j^2ux})\\ &= \lambda \int^{\infty}...
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1answer
78 views

$\varphi_{X+Y}(t)=\varphi_X(t) \cdot \varphi_Y(t)$, but X and Y are not independent

Consider $X,Y$ random variables with joint distribution: $$f_{X,Y}(x,y)=\begin{cases} \frac14\left[ 1+xy(x^2-y^2)\right] & |x|\leq 1,\;|y|\leq 1 \\ 0 & \text{otherwise} \end{cases}$$ ...
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1answer
42 views

calculate the integral $ \int_{-\infty}^{+\infty} \frac{e^{-itx}}{1 + t^2} dt $

It is possible to calculate the integral $ \int_{-\infty}^{+\infty} \frac{e^{-itx}}{1 + t^2} dt $ without using the residue theorem, nor fourier transforms
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1answer
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Use characteristic function of X. (X is a Laplace (0,1) distribution) to obtain characteristic function of the standar Cauchy distribution

Let X a r.v. with pdf: $\;f(x) = \tfrac{1}{2}e^{-|x|}$ (Laplace(0,1)) a) Calculate the characteristic function of X No problem. I do it. $\varphi_{X}(t)=\tfrac{1}{1+t{^2}}$ b) Use the previous ...
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51 views

Prove that $\cos^2t$ is a characteristic function

Prove that $\cos^2t$ is a characteristic function. I really do not understand how to prove this. I know that if it is a characteristic function, it must fulfill some properties. But so far in all ...
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0answers
29 views

Z = $Y$ $X_1$ + $(1-Y)$ $X_2$. where Y is a Bernoulli random variable. Calculate characteristic function of Z

Let Z = $Y$ $X_1$ + $(1-Y)$ $X_2$, where Y is a Bernoulli random variable. Y is independent of $X_1$ and $X_2$. Find the characteristic function of Z. It's posible calculate the characteristic ...
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1answer
28 views

Find the characteristic polynomial through the induction

$A = \begin{bmatrix} 0 & 0 & \dots & 0 & a_{0} \\ 1 & 0 & \dots & 0 & a_{1} \\ \ 0 & \ddots & \ddots & \vdots & \...
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2answers
30 views

How to calculate the limit of the characteristic functions $\chi_{[0,n]}$ and $\chi_{[-n,n]}$

Hi I am unsure of how you would evaluate the limit as $n\longrightarrow \infty$ of characteristic functions such as $\chi_{[0,n]}$ $\chi_{[-n,n]}$. Would their limits simply be $\lim_{n\to\infty} \...
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2answers
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Derivative of a function involving a characteristic function.

Consider the function $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ defined by $$f(x,y,z) = y^2z\chi_{(0,\infty)^3}.$$ I would like to find $$\frac{\partial^3f}{\partial x\,\partial y\,\partial z}.$$ ...
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If $X$ and $Y$ are iid with mean $0$ and variance $1$, and $T= \frac{X+Y}{\sqrt{ 2 }}$ has the same distribution, prove it is standard normal [closed]

Let $X$ and $Y$ be two i.i.d. random variables with mean $E[X]=E[Y]=0$ and variance $Var(X)=Var(Y)=1$. Let $T=\left( \frac{X+Y}{\sqrt{ 2 }} \right)$ and suppose that the distribution of $T$ is the ...
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1answer
36 views

Let X and Y two independent random variables with exponential distribution of parameter a>0. U = X+ Y and V = X- Y are not independent

Let X and Y two independent random variables with exponential distribution of parameter a>0. Proof using characteristic functions, that U = X+ Y and V = X- Y are not independent. 1) I calculate the ...
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0answers
25 views

Techniques for Fourier Transforms in Probability Theory

Question: What are the typical methods used to compute Fourier transforms for probability type integrals? There are so many different techniques for integration, but since probability integrals seem ...
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3answers
88 views

Finding the general formula for a sequence:

I have a sequence: $a_0 = 0;\ \ a_1 = 4;\ \ a_2 = 9; \ \ a_n = 4a_{n-1} - 5a_{n-2} + 2a_{n-3}$ I want to find the general formula.
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1answer
17 views

Finding minimal polynomial with given operator

Given the operator $T:\mathbb C_{\le n}[x]→\mathbb C_{\le n}[x]$ such that $T(p) = p' + p$ find the minimal polynomial. What I tried: I found the representing matrix $$A = \begin{pmatrix} 1 & ...
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1answer
53 views

Let $a, b, c ∈ R$, and let $M$ be the following matrix. Prove that $(a + c)^2 − 4ac ≥ 0$.

Let $a, b, c ∈ R$, and let $M$ = \begin{pmatrix} a & b \\ b & c \end{pmatrix} Prove that M has a real eigenvalue
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1answer
35 views

Largest eigenvalue of the Laplacian Matrix in an odd cycle

Problem: We have an odd cycle, $C_{2n+1}$, for $n \geq 1$, and the edges $e \in E\ $ have all one weights $w \in \{1\}^E$. Question: Denote the largest eigenvalue of the Laplacian matrix of this ...
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2answers
60 views

Compute $I = \int_{0}^{\infty} \frac{8}{\gamma^{4}} x^{3} e^{-\frac{2}{\gamma^{2}}x^{2}} e^{jxt} dx$

I want to evaluate the following integral: $$I = \int_{0}^{\infty} \frac{8}{\gamma^{4}} x^{3} e^{-\frac{2}{\gamma^{2}}x^{2}} e^{jxt} dx $$ Where $\gamma \in \mathbb{R}$ and $j = \sqrt{-1}$. The ...
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Compute $I(t) = \int_{0}^{\infty} \frac{1}{2^{k/2} \Gamma(\frac{k}{2})} x^{\frac{k}{2}-1}e^{-\frac{x}{2}} e^{jxt} dx$

I want to solve the following integral: $$ I(t) = \int_{0}^{\infty} \frac{1}{2^{k/2} \Gamma(\frac{k}{2})} x^{\frac{k}{2}-1}e^{-\frac{x}{2}} e^{jxt} dx $$ Where $j = \sqrt{-1}$, and $k \in \mathbb{N}...
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0answers
75 views

Merging with respect to bounded uniformly continuous functions in terms of characteristic functions

I would like to know if there are any results, where merging of probability measures in $R^n$ with respect to bounded uniformly continuous functions is deduced from some conditions on characteristic ...
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Moments of the distribution relating to smoothness of the characteristic function

It is known that if the nth moment of a random variable X exists, then its characteristic function is n times continuously differentiable. My intuition here is that we can think of the distribution ...
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1answer
28 views

Intuitions about positive definite functions

I am looking for further intuitions about positive definite functions, and have several related questions on this matter. I know this isn't the most specific question, but I find that speaking ...
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0answers
41 views

A question about lattice distribution and characteristic function

A random variable $X$ has a lattice distribution if the support of $X$ is $\{a+nb: n\in Z\}$. Let $X$ have characteristic function $\phi$. I have proved earlier that $X$ has a lattice distribution $\...
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1answer
39 views

Proving the Linearity of the Lebesgue Integral of Simple Functions

I don't understand a particular step in many proofs showing the linearity of the Lebesgue integral of simple functions. Consider the canonical decomposition of a simple function $\phi = \sum_{j=1}^{N}...
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1answer
29 views

How to show the following weak convergence using characteristic function

$X_i$ are $iid$, $EX_i = 0$, $EX_i ^2 = \sigma ^2$, $0<\sigma ^2<\infty$ Need to show: $\frac{\sum_{m=1}^n X_m}{\sqrt{\sum_{m=1}^n X_m ^2}}\xrightarrow{w} N(0,1)$ My attempt: Weak ...
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1answer
42 views

How to show the following weak convergence using characteristic functions

Suppose $g:R\rightarrow R$ has at least three bounded continuous derivatives and let $X_i$ be $iid$ and in $L^2$. Prove that: $\sqrt{n}[g(\overline{X_n}) - g(\mu)]\xrightarrow{w} N(0,g^{'}(\mu)^{2} ...
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Understanding the proof of inversion formula for density using characteristic function

The formula is: $f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\lambda x}\hat{f}(\lambda)d\lambda$ where $\hat{f}$ is the characteristic function, $f$ is continuous bounded on $R$ and both $f,...
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1answer
32 views

The steps behind finding the characteristic function of RV's under transformation

I have recently been introduced to the method to find the characteristic function of a random variable that stems from transformations of other random variables. Say, for example, $X, Y$~$\mathcal{N}(...
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0answers
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Characterization of Symmetric $\alpha$ Stable Distributions. [duplicate]

Question Show that the only symmetric $\alpha$ stable distributions with $0<\alpha\leq 2$ are $\phi_{\alpha}(t)=\exp(-|t|^\alpha)$ and their scaled versions (where $\phi$ denotes the ...
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1answer
55 views

Calculate the characteristic function of $S_n = X_1 + X_2 + . . . + X_n$

Let $X_1, X_2, . . . , X_n$ be mutually independent copies of X. Calculate the characteristic function of $S_n = X_1 + X_2 + . . . + X_n$. Determine how $S_n$ is distributed. I am trying to solve the ...
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1answer
17 views

Finding characteristic function and differentiate to get expectation

I was asked to find the characteristic function of a pdf and differentiate it to get the expectation. $p(x) = xe^{-x}$ for $x \ge 0$ I am doing this in the following way. Sorry that i don't know ...
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1answer
40 views

Poisson as a limit of the Binomial Characteristic Function

We are given $X_n\sim B(n,p_n)$ where $np_n\rightarrow\lambda$, and $\lambda>0$. The goal is to prove $X_n$ converges in distribution to Poisson($\lambda$) by use of characteristic functions. ...
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2answers
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Convolution of $\chi_{[0,1]}$ with itself [duplicate]

The characteristic function of a set $E$ is defined as follows: $\chi_{E}(x) :=1 \space \text{if} \space x\in E, \space \text{and} \space \chi_{E}(x) := 0 \space \text{if} \space x \notin E.$ Find a ...
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1answer
39 views

Inequality in characteristic function

Let $\phi$ be a characteristic function of random variable $X$. Prove that $1-|\phi(2u)|^2\leq 4(1-|\phi(u)|^2)$. I don't even have a clue how to start this.
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2answers
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Proving that if $tX_1 + sX_2 \stackrel{D}{=} \sqrt{t^2 + s^2}X$ then $X \stackrel{D}{=} N(0, \delta^2)$

Assume that for all $s, t \in \mathbb{R}$ the following property $$tX_1 + sX_2 \stackrel{D}{=} \sqrt{t^2 + s^2}X \tag{1}$$ is true. Moreover $X_1, X_2, X$ are i. i. d. My task is to prove that if $(...
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1answer
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Finding if $\phi(t) = \frac{\cos(t)}{1 + t^4}$ is a characteristic function

Let's consider a function: $$\phi(t) = \frac{\cos(t)}{1 + t^4} \tag{1}.$$ How can I check whether $(1)$ is a characteristic function? I tried using Polya's criterion. Unfortunately it doesn't work ...
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0answers
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General Central Limit Theorem for Binomial Random Variables

Question Let $(X_n)_{n\geq 1}$ be a sequence of arbitrary binomial random variables such that $EX_n\to \infty$ and $\text{Var}(X_n)/EX_n^2\to 0$ as $n\to \infty$. Then show that $$ Z_n=\frac{...
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1answer
39 views

Showing $ \mathbb{P}\Big( \frac{\Pi - \lambda}{\sqrt{\lambda}}\leq x \Big) $ [duplicate]

Let $\Pi$ be a random variable distributed by Poisson distribution with parameter $\lambda>0.$ Need to show that $$ \mathbb{P}\Big( \frac{\Pi - \lambda}{\sqrt{\lambda}}\leq x \Big) \rightarrow_{\...
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1answer
35 views

Showing $1-\Re \varphi(2t)\leq 4(1-\Re \varphi(t)) $

I need to show that $1-\Re \varphi(2t)\leq 4(1-\Re \varphi(t)) $ where $t \in \mathbb{R}$ for evry characteristic function $\varphi$. I know that if the random variable is symmetrical then the ...
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1answer
28 views

Finding characteristic function then density function is given

Random variable $\xi$ is distributed by symmetrical principle with density function $\frac{1}{2a} \mathcal{1}_{[-a,a]}(x),$ here $a>0$. I need to find characteristic function. I never seen ...
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1answer
45 views

Characteristic function of independent Poisson random variables

Let $X_{i}$ be independent Poisson distributed random variables with parameter $\lambda_{i} > 0$ for $i = 1,\ldots,n$. Now the joint distribution is given by \begin{equation*} \mathbb{P}\left(X_{...
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1answer
30 views

Characteristic function of normal distribution with complex parameter

Suppose that $X$ is a random variable that follows a normal distribution $N(0,\sigma^2)$. We know that its characteristic function can be computed as follows $$ \mathbb{E}[e^{i t X}]=e^{-\frac{1}{2}t^...
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1answer
68 views

If $X_n$ is Gamma $(n,\lambda)$ distributed then $(\lambda X_n -n)/\sqrt n\to N(0,1)$

Let $X_n$ be Gamma $(n,\lambda)$ distributed, and $Y_n = \dfrac{\lambda X_n -n}{\sqrt{n}}$. Show that $Y_n \rightarrow N(0,1)$. My idea to prove this is to use Lévys theorem with the ...
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1answer
56 views

characteristic function of $WX$ where $W$ and $X$ are independent standard normal random variables

Let $W$ and $X$ be independent random variables, both standard normal distributed. I have to show that for the characteristic function of $WX$ it holds that $\phi_{WX}(u) = \frac {1}{\sqrt{1+u^2}}...
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1answer
38 views

Proving that if $\varphi(t)$ is an infinitely divisible characteristic function then $|\varphi(t)|$ as well

Problem. I am given an infinitely divisible characteristic function $\varphi(t)$. My task it to prove that $|\varphi(t)|$ is infinitely divisible too. My attempt. Because $\varphi$ is infinitely ...
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1answer
45 views

Show that $\psi(t)=e^{\lambda (\varphi(t)-1)}$ is infinitely divisble for any characteristic function $\varphi$

I am given a function $$e^{\lambda(\varphi(t) -1)} \tag{1},$$ where $\varphi(t)$ is a characteristic function. I managed to show that $(1)$ is a characteristic function too. Now I am to show that $(1)$...
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0answers
43 views

$X_n$ ~ $\Gamma(n,n)$, find the limit in Law of $X_n$

I am investigating the following idea. Let $X_n$ ~ $\Gamma(n,n)$. I want to find the limit in Law of this random variable. I tried using Paul Levy theorem that says the following: If I find the ...
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2answers
42 views

Why $|e^{itx}| = 1$?

I'm studying some demonstrations of properties of characteristic function in which I have to use that $|e^{itx}| = 1$ but I don't understand it at all. Could you give a clue to demonstrate it?
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3answers
87 views

How to integrate moving Heaviside block?

I spent a lot of time figuring out how to integrate a convolution of a heaviside function with another heaviside function, but so far I couldn't find any closed form. $$\int_{-\infty}^{\infty} H\...
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0answers
87 views

Characteristic Function of Gamma Distributed Random Variables

I have the following characteristic function $$\sum_{m=0}^{\infty} \frac{(is)^m}{m!} \sigma_{m,k} \frac{\Gamma(\beta + m)}{\Gamma(\beta)},$$ where $i$ is the imaginary unit, $\beta>0$, $\Gamma(\...
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0answers
40 views

if a characteristic function is constant on a interval then is constant in all R.

I need help with this, please: If a characteristic function is constant equals 1 on a interval $-r\leq t\leq r$ then is constant in all $\mathbb{R}$