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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A basic question on characteristic function

Suppose I have two random variables $X$ and $Y$ for which characteristic functions are same. Let $F$ and $G$ be their distribution functions. I have to prove that $F$ and $G$ have the same set of ...
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0answers
16 views

Expected value and Differentiation of Characteristic function

Is there an example of random variable that has characteristic function to be differentiable at zero, but has no expected value?
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1answer
24 views

PDE $ u_{x}+u_{t}+f(x)*u=0$

How would I solve this pde using characteristic line? $u_{x}+u_{t}+f(x)u=0$---arbitrary function f $u(x,0)=u_{0}(x)$---$u_{0}$ can be any value $u(0,t)=\varphi(t)$---non-homogeneous where $u(x,t)\ge ...
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1answer
18 views

Calculation of characteristic functions of Levy processes

Let us say we have some Levy process $X_t$ and want to calculate its characteristic function, $E[e^{iuX_t}]$ for a certain value $u$. Is there a general procedure for this? I can imagine a way of ...
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1answer
20 views

Empirical characteristic function

The ecf is $\phi_n(\omega) = \frac{1}{n}\sum_{j=1}^ne^{iX_j\omega}$. I'm stuck on trying to see why the following is true $$|\phi_n(\omega)|^2 = \phi_n(\omega)\phi_n(-\omega)$$ Wouldn't this imply ...
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0answers
13 views

Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
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1answer
32 views

Limiting distribution of $X_n1(|X_n|\le 1-\frac{1}{n})+n1(|X_n|>1-\frac{1}{n})$ if $X_n\sim Unif(-1,1)$ and are iid.

Limiting distribution of $X_n1(|X_n|\le 1-\frac{1}{n})+n1(|X_n|>1-\frac{1}{n})$ if $X_n\sim Unif(-1,1)$ and are iid. From looking at the term, if $n$ goes to infinity, then $Y_n$ would be $X_n$ so ...
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3answers
30 views

Properties of the Characteristic/Indicator Function

Let $B_1,B_2,...$ be a countable family of disjoint subsets of $\Bbb R^d$. For any set $E \in \Bbb R^d$, let $\chi_E (x)=1$ if $x \in E$ and $\chi_E (x)=0$ otherwise. Is it true that $\chi_{\bigcup ...
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26 views

Inverting a difficult characteristic function.

Let $X$ be a random variable with characteristic function $$\phi_X(w)= \frac{1}{1-\frac{iw}{\lambda}e^{(\lambda-iw)\eta }}$$ where $\lambda$ and $\eta$ are constants. What is the pdf of $X$?
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1answer
32 views

Why can I use the Riemann-integral here?

Let $Z\sim\mathcal{N}(0,1)$ (i.e. a random variable which distribution is the standard normal distribution). Determine the characteristical function of $Z$. It is $\mathbb{P}_Z=f\lambda$ ...
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1answer
43 views

Why is $\int e^{itx}\, d\mathbb{P}_X=\mathbb{E}(e^{itX})$?

In our reading we first defined the characteristical function of a probability mesaure as follows: Let $\mu$ be a probability measure on $(\mathbb{R},\mathcal{B})$. The Fourier transform ...
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0answers
26 views

Show that if X has a density f such that f’ exists and is integrable?

Show that if $X$ has a density $f$ such that $f'$ exists and is integrable, then its characteristic function has the property : $\phi(t)=ο(t^{-1} )$ as $t\to \infty$. Hint: If $X$ has a density ...
3
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1answer
146 views

Exercise on Conditional Expectation of Jointly Gaussian Random Variables

I am trying to solve the following exercise from my professor's notes on conditional expectation: Let $x: \Omega \rightarrow \mathbb{R}^n$, $x \in G(0, Q_x)$, $Q_x = Q_x^T>0$, $y: \Omega ...
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1answer
55 views

the characteristic function of this distribution is equal to 0 everywhere except at the origin, mistake?

I wanted to compute the characteristic function of the distribution in question here: How to multiply a standard normal RV times a uniform{-1.1} RV? Let $X$ be standard $N(0,1)$, $Y$ be Uniform ...
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0answers
16 views

Charateristic function evaluation

I have a signal given by the following equation: $y_k = X_k S_0 + \sum_{l=0 \& l\neq k}^{N-1}S_{l-k}+n_k$ where $X_k$ are independent and identically distributed random variables. $n_k$ is a ...
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0answers
63 views

Does the Riemann zeta function tell us about the order theoretic properties of the natural numbers?

The classical Möbius function $\mu(n)$ fulfills the multiplicative inversion formula, e.g. see this thread. Now I see in the theory of posets, they generalize the concept of that function, see ...
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0answers
34 views

I want to show $\phi_{X}(a_{1},a_{2},\cdots,a_{n})=\prod_{i=1}^{n}\phi_{X_{i}}(a_{i})$

Let $n \in \mathbb N$ and $X$ be an $\mathbb R^n$ valued random variable on $(\Omega ,\mathcal F,P)$ Define its characteristic function to be $$\phi_{X}(a)=E(e^{i\langle X,a\rangle})$$ where $a \in ...
3
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1answer
61 views

Stable law and Levy distribution

A PDF (probability density function) f(x) is called a stable law if $f(y)=b\int_{-\infty}^{\infty}dx f(by-x)f(x)$ under appropriate values of b. Rewrite this equation in terms of characteristic ...
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1answer
97 views

Characteristic function of an integer-valued distribution, inversion formula

I am working on the following: Show that if $\varphi$ the characteristic function of an integer-valued distribution then \begin{align*} \mathbb P(X = k) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{-itk} ...
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0answers
33 views

Characteristic function of a exponential random variable, problems with complex integral.

I tried to compute the characteristic function of a random variable, which is exponential distributed with parameter $\lambda$: \begin{align*} \varphi(t) &= \mathbb E[e^{itX}] = ...
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1answer
50 views

A function of $u(0,1)$ random variables converging weakly to an exponential

This is a review problem for my final exam: Let $(X_{n})_{n\geq 1}$ be an i.i.d. sequence of random variables with $X_{i} \sim U(0,1)$. Let $M_{n}=\max_{1\leq i \leq n}X_{i}$. Show that $n(1-M_{n})$ ...
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1answer
21 views

Find the standard representation of a function and the Lebesgue integral

Find the standard representation of the function f defined by f(x)=[x] for −1≤x≤3, f(x)=0 otherwise. determine the integral R of fdu I came across this question while studying and began to attempt ...
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1answer
97 views

Characteristic function of Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal ...
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0answers
29 views

characterization of characteristic functions (Bochner Theorem Proof?) Simple case.

Prove the following theorem: Let $\phi: \Bbb R \to \Bbb C$. $\phi$ is the characteristic function of a real random variable $X:\Omega \to \Bbb R$ if and only if $\phi(0)=1$ $\phi$ is uniformly ...
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0answers
21 views

Proving the characteristic equation

Consider the recurrence relation: $a_n = \alpha_1 a_{n-1}+\alpha_2 a_{n-2}+...+\alpha_k a_{n-k} ,$ where $\alpha_1 , \alpha_2 , ... \alpha_n $ are constants. 1) Prove that if $b$ is a non-zero ...
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1answer
45 views

Characteristic function say something about the expectation and variance [closed]

Show that if $\lim_{t \downarrow 0} (\varphi(t) -1) / t^2 = c > -\infty$ then $EX = 0$ and $E|X|^2 = -2c < \infty$. In particular, if $\varphi(t) = 1 + o(t^2)$, then $\varphi(t) \equiv 1$. Where ...
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1answer
46 views

Convergence in distribution of independent and uniform r.v.'s

This is the text of the problem: Let $\left(X_{j}\right)_{j\ \geq\ 1}$ be independent and let $X_{j}$ have the uniform distribution on $\left(-j,j\right)$. Show that $\lim_{n \to \infty}{S_{n} \over ...
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1answer
47 views

the density of the sum of $n$ random variables with uniform distribution on $(-1,1)$

Let $X_n$ be an iid sequence of random variable with uniform distribution on $(-1,1)$. Using characteristic functions prove that $X_1+X_2+...+X_n$ has density $$f(x)= \frac{1}{\pi} ...
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1answer
59 views

Linear Transformations $T$ and $S$ and their Characteristic Polynomials

My friends and I cannot figure out this proof. We have part (a) done, but weren't not quite getting part(b). We think we need a change-of-basis equation. Any advice?
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1answer
64 views

$\mathscr{B}$-matrix of T and Characteristic Polynomial

I'm having a difficult time trying to figure out this proof problem. Any advice on first steps? Let A be an $n\times n$ matrix satisfying the matrix equation $A^{n} + ...
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1answer
34 views

characteristic function characterize the distribution

Theorem: Let $\phi(t)=\int{e^{itX}dF_X}$ be a characteristic function of a random variable $X$. Then $\displaystyle \lim_{T \to \infty}\int_{-T}^{T}{{\frac{e^{-ita}-e^{-itb}}{it}}\phi(t)dt}=P(X\in ...
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1answer
42 views

What would be the simplified form of this expression?

I'm working on a Homework problem involving Convergence of Random variables and I've arrived at an expression which looks like follows: $$ M_{X_n}(ju)= ...
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1answer
44 views

Show that $\frac{1}{n}\sum_{j=1}^{n}X_{j}$ is Cauchy distributed when the $X_{i}$ are all Cauchy

Let $X_{1}, \cdots, X_{n}$ be i.i.d. Cauchy random variables with parameters $\alpha=0$ and $\beta=1$. (That is, their density is $f(x)=\frac{1}{\pi\,(1+x^{2})}$, $-\infty < x < \infty$.) Show ...
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46 views

Possible values of characteristic functions (Fourier transforms)

Can a characteristic function $\varphi_{X}(u)$ from probability theory (the Fourier transform of a probability measure) ever equal zero for either any value of $x$ or any value of $u$? This has been ...
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1answer
56 views

How does what I did imply that $X$ is Normal $N(0,1)$?

Let $X$, $Y$ be i.i.d, that $X+Y$ and $X-Y$ are independent, and that $\varphi_{X}(2u)=(\varphi_{X}(u))^{3}\varphi_{X}(-u)$. Also, let $E\{X\}=0$ and $E\{X^{2}\}=1$. Show that $X$ is Normal $N(0,1)$. ...
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1answer
31 views

Show that $|1-\varphi_X (u)|\leq E\{ |uX| \}$

Show that $|1-\exp\{ix\}|^{2}=2(1-\cos x) \leq x^{2}$ for all $x \in \mathbb{R}$. Use this to show that $|1-\varphi_X(u)|\leq E\{|uX|\}$, where $\varphi_X(u) =E\{\exp(i\langle u,X\rangle)\}$ is the ...
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1answer
67 views

Let X, Y be i.i.d, X+Y and X-Y independent, show that the characteristic function E{exp(i<2u,x>)}= …

Let $X$ and $Y$ be i.i.d. Suppose further that $X+Y$ and $X-Y$ are independent. Show that $\varphi_{X}(2u)=(\varphi_{X}(u))^{3}\varphi_{X}(-u)$. What I tried to do was work backwards, starting with ...
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0answers
43 views

For X, Y real valued and independent, and X and X+Y having the same distribution, Y=0 a.s.

Let X, Y be real valued and independent. Suppose X and X+Y have the same distribution. Show that Y is a constant r.v. equal to 0 almost surely. Here's what I have so far: By the uniqueness of ...
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1answer
37 views

What is the meaning of $1_{a>b}$?

What would this mean: $1_{a>b}$ .. Based on the context, it could mean "$1$ if $a>b$ else $0$", but it's the first time I see it so help would be appreciated.
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0answers
34 views

Limit of the expectation of the sum

Show that for $g(t)= E \left\{\sum_{n=3}^{\infty}\frac{(iut)^{n}}{n!}\right\}$ that $\lim_{t \to 0} \frac{|g(t)|}{t} =0$. I think I should bound it and then use LDCT, but I'm having trouble doing ...
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1answer
28 views

Even numbered moments of N(0,1) using characteristic functions

Let $X$ be $N(0,1)$. Show that $E\{X^{2n+1}\}=0$ (Easy - calculate it directly using the definition of expectation, and you're taking the integral of an odd function over a symmetric interval, so =0), ...
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62 views

Characteristic function of an r.v. with finite variance and zero mean.

Suppose $E{|X|^{2}}<\infty$ and $E{X}=0$. Show that Var(X)$= \sigma^{2}<\infty$ (done), and that $\varphi_{X}(u)=1-\frac{1}{2}u^{2}\sigma^{2}+o(u^{2})$ (what I can't figure out how to find, ...
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4answers
93 views

Showing the expectation of the third moment of a sum = the sum of the expectation of the third moment

Let $X_{1},\cdots,X_{n}$ be independent, each with mean 0, and each with finite third moments. Show that $E\left\{\left( \sum_{i=1}^{n}X_{i}\right)^{3}\right\} = \sum_{i=1}^{n}E\left\{ X_{i}^{3} ...
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175 views

Characteristic Function of a Double Exponential (Laplace) Distribution

Let X have the double exponential (or Laplace) distribution with $\alpha =0$, $\beta = 1$: $f_{X}(x)=\frac{1}{2}e^{-|x|}$, $-\infty < x < \infty$. Show that $\varphi _{X}(u)=\frac{1}{1+u^{2}}$. ...
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0answers
80 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...
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1answer
64 views

Can the characteristic function of a multivariate normal distribution be extended from a neighborhood of the origin?

Let $x$ be a scalar random variable. There is a theorem that states that if $E[\exp(ixs)]= \exp\Big( i{s}\mu - \tfrac{1}{2} {\sigma^2s^2} \Big)$ for some neighborhood around the origin (i.e. ...
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2answers
40 views

Finding the characteristic ODE from a nonlinear PDE

I am studying for a PDE exam on Tuesday, and I am getting pretty confused about one specific type of problem and I am thinking that perhaps I am misinterpreting the correct procedure to follow. The ...
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1answer
83 views

Prove that $ \mathsf{E}[g(X)] = \int_{- \infty}^{\infty} G(t) \varphi(t) \, d{t} $.

Problem Let $ X $ be a real-valued random variable with characteristic function $ \varphi $. Suppose that $ g: \mathbb{R} \to \mathbb{R} $ satisfies $$ \forall x \in \mathbb{R}: \quad g(x) = ...
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1answer
132 views

Characteristic function

Question: Let $X_1$ and $X_2$ denote independent real-valued random variables with distribution functions $F_1$, $F_2$, and characteristic functions $\varphi_1$, $\varphi_2$, respectively. Let Y ...
2
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2answers
116 views

Characteristic function of $p(x) = \frac{1}{2} e^{-|x|}$, $-\infty < x < \infty$

Let X denote a real-valued random variable with an absolutely continuous distribution with density function $p(x) = \frac{1}{2} e^{-|x|}$, $-\infty < x < \infty$. Find the characteristic ...