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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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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
27 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
15 views

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
46 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
16 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 ...
2
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1answer
31 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
61 views

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
36 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
32 views

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
59 views

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
29 views

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
35 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
31 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
24 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
29 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
16 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
50 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 ...
2
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1answer
42 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
33 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
41 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
40 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
40 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
84 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\...
2
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0answers
51 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
32 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 $\mathbb{R}$
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1answer
44 views

A nice inequality about characteristic function [closed]

I need help with this please. $\psi$ is characteristic function of a probability measure, then: $ |\psi(t)-\psi(s)|^2\leq 2|1-\psi(t-s)|$
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0answers
55 views

Show that stochastic integral w.r.t. Brownian motion is normal distributed

I want to show the following claim: Let $B$ be a one-dimensional Brownian motion and let $$I(\phi):=\int_0^1 \phi(s) \text{d}B_s.$$ Show that $\mathbb{E}(I(\phi))=0$ and $\mathbb{V}(I(\phi))=\int_0^1(...
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0answers
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Show that $\frac{d}{dt} E\Big[\big |\phi_X(tZ)\big|^2\Big ]=2 E \left[ Z \phi_X(-t Z) \phi_X'(t Z) \right] $.

Let $Z$ be standard normal. Is the following sequence of steps correct? \begin{align} \frac{d}{dt} E \Big[ \big|\phi_X(tZ)\big|^2 \Big] &= E \left[ \frac{d}{dt} |\phi_X(tZ)|^2 \right] \\ &...
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0answers
38 views

Showing that the Distribution of a Random Variable Must be Standard Normal using Characteristic Functions [duplicate]

Question Let $X$ and $Y$ be i.i.d with means $0$ and variances $1$. Let $\phi(t)$ be their common characteristic function and suppose that $X+Y$ and $X-Y$ are independent. Show that $\phi(2t)=\phi(...
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2answers
35 views

Showing that the ratio of two standard independent normals is a Cauchy using Characteristic Functions

Question Let $X$ and $Y$ be independent standard normals. Use characteristic functions to find the distribution of $X/Y$. My attempt We will attempt to show that $Ee^{itX/Y}=e^{-|t|}$ (the c.f. ...
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1answer
90 views

Question involving Characteristic Functions and the Existence of a Distribution

Question Is it possible for $X$, $Y$ and $Z$ to have the same distribution and satisfy $X=U(Y+Z)$ where $U$ is uniform on $[0,1]$ and $Y$, $Z$ are independent of $U$ and of one another? The above ...
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1answer
38 views

If $\phi$ is a characteristic function, then $1-|\phi(2t)|\leq 8\{1-|\phi(t)|\}$

Question If $\phi$ is a characteristic function, show that $\text{Re}\{1-\phi(t)\}\geq \frac{1}{4}\text{Re}(1-\phi(2t))$ and deduce that $1-|\phi(2t)|\leq 8\{1-|\phi(t)|\}$. My attempt I have ...
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1answer
66 views

Show that $\int_{\mathbb{R}} |\mathbb{E}(e^{i \xi X})|^2 \, d\xi < \infty$ implies $P(X=x)=0, \forall x\in\mathbb{R}$. [duplicate]

Let $X$ be a real-valued random variable with $\varphi_X\in L^2(\mathbb{R})$, where $\varphi_X$ denotes the characteristic function of $X$. Prove that $P(X=x)=0, \forall x\in\mathbb{R}$. Since $\...
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0answers
39 views

Prove that $\phi(t)= \frac{1}{2} + \frac{e^{t^2}}{4e^{t^2} - 2}$ defines characteristic function

Let $\phi : \mathbb{R} \rightarrow \mathbb{R}$ be given by $$\phi(t) = \frac{1}{2} + \frac{e^{t^2}}{4e^{t^2} - 2}$$ Prove that $\phi$ is a characteristic function. My attempt: I know that there ...
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1answer
38 views

Prove that the given RV is normal

This is an old qualifying exam problem of probability theory. Let $X,Y$ be two iid random variables with mean zero and variance 1. Suppose that $X,Y,\frac{X+Y}{\sqrt{2}}$ are all identically ...
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2answers
50 views

Proof by characteristic functions that $X+Y$ and $2X$ are identically distributed

The exercise states that X,Y iid and we know that X+Y has Cauchy distribution. And they require to prove that 2X has also Cauchy distribution. Let me put it straight, I dont think I understand it ...
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0answers
68 views

Characteristic function equal to 1

Is the fact that the characteristic function of $X$ is equal to 1 enough to say that $X=0$ almost surely? Is this the correct proof? The characteristic function is determined uniquely. The random ...
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1answer
27 views

How to show Sub-independent Random Variables are uncorrelated.

I want to prove the following: If two RVs $X, Y$ are sub-independent, i.e., $\phi_{X+Y}(t) = \phi_X(t)\phi_Y(t), t\in\mathbb{R}$ then $X, Y$are uncorrelated. Keep $Cov(X,Y) = E(XY)-E(X)E(Y) = 0$ in ...
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0answers
51 views

If $\mathbb{E} e^{it(X+Y)} = \mathbb{E} e^{itX} \mathbb{E} e^{itY}$ then $X$ and $Y$ are uncorrelated

Let $X$ and $Y$ be random variables such that $\mathbb{E} e^{it(X+Y)} = \mathbb{E} e^{itX} \mathbb{E} e^{itY}$ and covariance exists. I want to show that they are uncorrelated, i.e., $\mathbb{E}(X -\...
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0answers
48 views

Characteristic function of infinitely divisible measure has no zero.

I want to show that the characteristic function of an infinitely divisible measure $P$ has no zero, directly by using the fact that for all $n\in\mathbb{N}$, there is a measure $P_n$, such that: $$\...
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0answers
44 views

Characteristic function and point masses

Let $X$ be some random variable with characteristic function $\hat{X}$. Show that $X$ has no point masses if and only if $$ \lim_{T\to +\infty}\frac1{2T}\int_{-T}^T\left\lvert \hat{X}(t)\right\...
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1answer
39 views

Functions of Characteristic Functions

Suppose that $\phi(t)$ is the characteristic function of a random variable. Show that the functions $\phi^2(t)$ and $|\phi(t)|^2$ are also characteristic functions. Attempt: So since $\phi(t)$ is a ...
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1answer
32 views

Why is it necessary X and Y beeing bounded when having same moments to be equal

Let $X,Y$ random variables in $[0,1]$ with $E(X^n)=E(Y^n) \,\forall n\in \mathbb N$. I want to show $X\overset{d}{=} Y$. $$E(e^{itX})=\int\sum \frac{(itx)^k}{k!}dF_X\overset{dom. conv.}{=}\sum\int\...
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1answer
16 views

Multivariable calculus discontinuities question

Question: If D is the set of discontinuities of $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$, show that the set of discontinuities of $f_{\Phi_{A}}$ is contained in $D \bigcup \partial A$. So, $f_{\Phi_{...
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1answer
54 views

Fourier uniqueness theorem

I am reading Theorem 26.2 in the book "Probability and Measure" (2nd ed.) by Patrick Billingsley. In the proof of that theorem, the author claims that: Fact Let $\mu$ and $\nu$ be probability ...
2
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1answer
62 views

characteristic function and distribution completely determined by moments

Let $X$ be a real valued random variable. My textbook states that if the moment generating function $E[e^{sX}]$ is finite in a neighborhood of zero, the distribution of $X$ is determined completely by ...
2
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0answers
30 views

Continuity of a Characteristic function in real-analysis

Let $F \subseteq [0,1]$ be a closed set with empty interior such that $m(F)>0$ Prove that the characteristic function $χ_F$ denfined on $[0,1]$ satisfies the following property: for every set $A \...
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1answer
38 views

Poisson and convergence in distribution

Let $X_1,X_2,...$ be iid with mean $\mu$ and variance $\sigma^2$. Let $N_{\lambda}$ be poisson($\lambda$) independent of the $X_i$'s. (a) Find the limit in distribution as $\lambda \rightarrow \infty$...
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1answer
30 views

characteristic function of a clopen set continuous?

Let $X$ be a compact Hausdorff space, $C(X)$ the continuous functions complex valued on $X$ and $U\subset X$ clopen. Is then the characteristic function $1_U\in C(X)$? $1_U:X\to\mathbb{C}$ is defined ...
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2answers
44 views

use indicator function to prove independence

I would like to know why for $\forall A \in \mathcal{F}$ $$E[1_{A}e^{itX}] = P(A)E[e^{itX}]$$ will imply the independence between $X$ and $\mathcal{F}$. Thanks in advance!