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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1answer
26 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
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0answers
30 views

Measure of $\chi_\mathbb{Q}(x)$?

$\chi_\mathbb{Q}(x) = 1$ if $x \in \mathbb{Q}, 0$ otherwise. Well $\chi_\mathbb{Q}(x)$ is a measurable function if $\mathbb{Q}$ is a measurable set. $\mathbb{Q}$ is a measuable set under the Borel ...
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0answers
18 views

Convergence of the empirical characteristic function

I have to prove that the empirical characteristic function of a sample of i.i.d. random values converges in distribution against the normal distribution. I think I have to use the Multidimensional ...
1
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1answer
49 views

Rewriting a double integral with complex exponential function

Why can we write $$ \begin{align} I_T &= \int_\mathbb{R}\int_{-T}^{T}\frac{e^{-ita}-e^{-itb}}{it}e^{itx}dtdF(x)\\ &= \int_\mathbb{R}\left[\int_{-T}^{T}\frac{\sin(t(x-a))}{t}dt - ...
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0answers
61 views

Probability:questions on characteristic functions

A well-known example to show that two random variables whose marginal distributions are normal, do not need necessarily be jointly normal is achieved by letting $X, Y $ have the following joint ...
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1answer
36 views

Prove if $E$ is a Lebesgue measurable set, there exists a continuous function $f$ differing from $\chi_{E}$ on a set of measure $< \epsilon$?

I am reviewing my analysis notes, and I don't really understand the proof given by my professor. He first proved if $E$ is a Lebesgue measurable set and $\epsilon > 0$, then there is an open set ...
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0answers
29 views

Integrating characteristic functions.

I need to find the radon transform of the following function. But I got stuck in finding this integral. Let $\chi$ be given by $$\chi(t) = \begin{cases} 1 & |t|< 1/2 \\ 0 & ...
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0answers
24 views

$X\sim\mathcal N(0,1)$, Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd?

If $X\sim\mathcal N(0,1)$ Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd ? ($\Phi_X^{(j)}(0):j^{th}$ derivative of the characteristic function of the r.v. $X$) We computed ...
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1answer
41 views

How to integrate $\int\mathbf 1_{(-\frac12,\frac12)}(z-w)\mathbf 1_{(-\frac12,\frac12)}(w)dw$

How to integrate $\displaystyle\int\mathbf 1_{(-\frac12,\frac12)}(z-w)\mathbf 1_{(-\frac12,\frac12)}(w)dw$ ? The integral should give a function of $z$, but I don't know how to compute. ...
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0answers
32 views

characteristic function of logarithm of random variable

If I know the characteristic function $\phi_X(t)$ of a random variable $X>0$, how can I write the characteristic function $\phi_Y(t)$ of $Y=\log(X)$? I know that $\phi_X(t)=E[e^{itX}]$ and ...
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1answer
59 views

Show that Z cannot be turned into a vector space over any field. [duplicate]

Show that Z cannot be turned into a vector space over any field. So, we have 2 cases here. Case 1:lets suppose the charF=P, n does not equal 0, then (1+1+...+1)n=1n+1n+...+1n=n+n+...+n=pn=wchich ...
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1answer
26 views

Convergence of $\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$ for $u_n \to u$?

Let $u_n \to u$ in $L^2(\Omega)$ and let $I$ be an bounded interval. Does it follow that $$\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$$ at least for a subsequence of $u_{n_j}$ ...
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1answer
16 views

characteristic equation in pde

In the PDE: $ yU_y-xU_x=1$ how did the characteristics become $dx\over -x$=$dy \over y$ =$du \over 1$.Can someone please expalin how these charactristic equations were obtained
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1answer
58 views

Characteristic Function Inversion

I am studying the relationship / bijection between characteristic functions and CDFs. In particular, given a characteristic function $\phi$ it is posible to recover the cumulative density function ...
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0answers
20 views

Finding Characteristic Exponents for $x^2 (x-1)^2 y'' + 4 (x-1)y' - 4x^2 y = 0$

I've found that the only regular singular point of this differential equation: $$x^2 (x-1)^2 y'' + 4 (x-1)y' - 4x^2 y = 0$$ is $x = 1$. How do I determine the characteristic exponents for it?
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1answer
136 views

Continuous probability distribution with no first moment but the characteristic function is differentiable

I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere. Cauchy distributions do not ...
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0answers
42 views

Characteristic function of compound Poisson process

It is widely known that the characteristic function of a compound Poisson process is $$ \phi_X(u) = \exp \left(t\lambda \int_{\mathbb{R}} (e^{iux}-1) F(dx) \right). $$ But if I try to derive it via ...
3
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1answer
75 views

Characteristic function under risk neutral measure

I am trying to derive a characteristic function (in Levy-Khintchine form) of a compound Poisson process $X_T$ under a risk neutral measure $\mathbb{Q}$, using the Esscher transfrom to change the ...
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4answers
100 views

Prove the matrix satisfies the equation $A^2 -4A-5I=0$ [closed]

How to prove that $$ A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} $$ satisfies the equation $A^2 -4A-5I=0$?
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1answer
25 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
44 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
26 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
16 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 ...
2
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1answer
38 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
47 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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1answer
35 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
48 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
232 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
65 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
64 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
38 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
69 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 ...
2
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1answer
106 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
40 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
51 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
22 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
152 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
37 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
52 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 ...
2
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1answer
48 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 ...
3
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1answer
53 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
67 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} + ...
1
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1answer
39 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)= ...
1
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1answer
47 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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0answers
51 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 ...