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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11 views

characteristic function upper bound and uniformly continuous.

Let $X$ be a random variable and let $\phi$ be its characteristic function. Let $A$ be a nonnegative constant and consider the following inequality $$ |\phi(t)-\phi(s)| \leq \sqrt{A|1-\phi(t-s)|}. $$ ...
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17 views

Find the Fourier transform of the functions [on hold]

I was wondering how to find the fourier transform of functions including the characteristic function. For example, how do I find the FT of: $$f(x)=\chi_{[-1,1]^d}(x), x\in \mathbb{R^d}$$ or ...
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64 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$? Let $\varphi_X(t)$ be ...
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1answer
32 views

Integral from $e^{-itx}$ over $\mathbb R$

As a part of my task considering characteristic functions I have to compute $\int_{\mathbb R} e^{-itx}dx$ The result i get is $\frac{1}{-it}e^{-itx}|^{\infty}_{-\infty}$, but I don't really know ...
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1answer
36 views

Is $ \frac{2}{1+e^{t^2}} $ a characteristic function?

I'm trying to establish whether the following is a characteristic function of some random variable: $$ \phi(t) = \frac{2}{1+ e^{t^2}} .$$ It satisfies all basic characteristic function properties, ...
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22 views

Characteristic functions and conditional distributions?

Say X and Y are random variables and we're interested in the conditional distribution of X given Y, can we make this calculation using only characteristic functions in a straightforward manner? If so ...
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35 views

An inequality with a characteristic function

It's my first question here, hi. In fact, it derives from my probability theory homework, which appears to be unusually difficult (or I just don't see something): Suppose $X$ is a real valued random ...
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0answers
11 views

Probability - Characterizing goodness of moment matching method.

I have a question about how to characterize the goodness of approximating a distribution using its moments. Suppose I have a probability density function $p(x)$ (e.g., normal distribution), and I am ...
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1answer
56 views

Dealing with Partial Differential Equations and Burger's equation

The problem is: consider Burger's equation, $$u_t +uu_x = 0 $$ $$ u(x,0) = f(x) $$ Where $$f(x) = \begin{cases} 1 - |x-2| &\mbox{if}\,\, 1\leq x \leq3, \\ 0 ...
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3answers
18 views

Characteristic function of a square of normally distributed RV

Let's assume that $ X \sim \mathcal{N}(0,1) $. I'm supposed to compute the characteristic function of $ X^2 $. As far as I got is that the density of $ X^2 $ is $ g(y) = \frac{1}{2\sqrt{2\pi}} ...
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1answer
27 views

Characteristic function of a product of random variables

I am facing the following problem. Let $X,Y$ be independent random variables with standard normal distribution. Find the characteristic function of a variable $ XY $. I have found some information, ...
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1answer
45 views

Is $\gamma(t)=\cos^2(t)$ a characteristic function?

Is $\gamma(t)=\cos^2(t)$ a characteristic function? I don't know how can I show that $\gamma(t)=\cos^2(t)$ is a positive function.
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10 views

Differences between the a.e continuous of function $\in \mathbb{R}^n$ in the sense of Peano Jordan and Lebesgue

Proves or disproves: (A) The indicator function $\chi_S$ of a bounded set $S \in \mathbb{R}^n$ is almost everywhere continuous in the sense of Peano Jordan $\Leftrightarrow$ is almost everywhere ...
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0answers
26 views

Solution to this non linear Equation

Hi all I have this equation $$ (y-\beta+2\beta e^{yd})(y+\beta -2 \beta e^{-yd})+\beta^2=0 $$ Solving for y, I have been told that the solution is y = 0 repeated roots. I can see why (from ...
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3answers
30 views

Symmetric Distribution of Random Variable

Prove: Let $X$ and $Y$ be random variables with the same distribution. If $X$ and $Y$ take only two values​​, then $X - Y$ are symmetrically distributed around zero. Note: 1 - You can use ...
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1answer
57 views

uniform convergence of characteristic functions

Assume that a sequence of probability measures $\mu_n$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Prove that $\phi_n$ ...
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1answer
32 views

characteristic function differentiation

Let $\mu$ be a probability measure on $\mathbb{R}$. Then the characteristic function is: $$ \varphi: \mathbb{R} \rightarrow \mathbb{C} \;\;\ \varphi(t):=i\int_\mathbb{R} e^{itx}d\mu(x) $$ Prove with ...
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1answer
26 views

Imaginary solutions of a recurrence relation

How to solve this recurrence relation using characteristic equation and imaginary numbers? We have $a_0 = 0$ and $a_1 = 1$ , and for all $j\in\mathbb N$: $$a_{j+2} = 6a_{j+1} - 10a_j$$ I would ...
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3answers
46 views

Solve recurrence relation problem

This is a recursion problem that I am stuck at. I need to use the characteristic equation. Let $a_0, a_1, a_2, . . .$ be defined by $a_0 = 5, a_1 = 0$, and $a_{n+2} = a_{n+1} + 6a_n$ for $n \ge 0$. ...
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0answers
18 views

Characteristic function of truncated Cauchy distribution

Truncated Cauchy distribution (with a symmetric truncation $-X\le x\le X$) has the density $$f(x)=\frac{1}{2\arctan(X)}\frac{1}{1+x^2}$$ What is the characteristic function of this r.v.? Thank's! ...
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23 views

recognize the distribution corresponding to this characteristic function

The characteristic function of a random variable X is given as $$\frac{3+cos(t)+cos(2t)}{5} $$, what is the distribution of X? I was thinking of the discrete random variable X=,0,1,2 with mass ...
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0answers
21 views

Transform Characteristic Function to distribution Function

I was wondering how we can calculate the distribution function of this characteristic function, $$C(t)= \frac{3+\cos t + \cos2t}{3}$$
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0answers
11 views

Order Statistics interval sizes

Suppose an i.i.d. sample of size $n \geq 2$ drawn from a known distribution with density $g$. Let us note the associated order statistics as $(X_{(1)}, \ldots ,X_{(n)})$. I am interested in the number ...
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1answer
37 views

Show that two sums have the same distribution?

I have not been able to show that the following two stochastic variables have the same distribution. My question is as follows: Let $$ X_1, X_2,..., X_n $$ be independent and identically ...
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2answers
43 views

On the characteristics function of smooth compactly supported distributions

My question is concerned with the Fourier transform of a density function of a continuous random variable (or characteristics function). In a book of Kim Lai Chung with the title "A course in ...
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0answers
21 views

By an example, show that..

Let $X_1$ ,$X_2$ be two r.v.'s with joint and marginal ch.f.'s $\phi_{X_1,X_2}$, $\phi_{X_1}$ and $\phi_{X_2}$. By an example, show that $\phi_{X_1,X_2} (t,t) = \phi_{X_1} (t) \phi_{X_2} (t)$ ...
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0answers
43 views

Clarification of Proof on Kac's Theorem for Characteristic Functions

There is a proof given here that I don't really understand, and was hoping someone more competent could explain it in some more detail: Moment generating functions/ Characteristic functions of $X,Y$ ...
2
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1answer
59 views

A linear combination of characteristic functions is a characteristic function?

Let $\phi_k(t)$ be the characteristic function of a random variable $X_k$, $k = 1,2,\dots$. Consider a set of positive real numbers $\{p_1, p_2, \dots \}$, take a function: $$\phi(t) = ...
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0answers
31 views

Recovering pmf from characteristic function

I'm having some trouble trying to recover the probability mass function of a discrete random variable from its characteristic function. I have seen that some continuous cases, you can recognize that ...
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1answer
73 views

Computing the characteristic function of a normal random vector

The characteristic function of a random vector $\boldsymbol{X}$ is $\varphi_{\boldsymbol{X}}(\boldsymbol{t}) =E[e^{i\boldsymbol{t}'\boldsymbol{X}}] $ Now suppose that $\boldsymbol{X} \in ...
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37 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
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2answers
76 views

How to show that $\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$

Let $x_n$ be a sequence of reals. Show that $$\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$$ Since the weak convergence is equivalent to pointwise convergence of characteristic functions ...
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2answers
73 views

Is $\exp(-2\sin^2t)$ a characteristic function?

Is $\exp(-2\sin^2t)$ the characteristic function of some random variable?
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1answer
41 views

Positive-definite + continuous at 0 $\Rightarrow$ continuous?

Let $F$ be a functional from $L_2(\mathbb{R})$ to $\mathbb{C}$ that is positive-definite*. We also know that $F$ is continuous at $0$. Can we deduce that $F$ is continuous over $L_2(\mathbb{R})$? ...
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0answers
41 views

Integral of Normal Distribution with imaginary unit

Hi I need some help with the following integral. $$ \int_{-\infty}^{\infty} \operatorname{e}^{itx} \cdot \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \operatorname{e}^{\frac{-(x - \mu)^2}{2\sigma^2}} \mathrm ...
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2answers
36 views

characteristic function of $\sum_i^N X_i$, $N$ is a Poisson distribution

I have a series of $X_i$ random variables, identically and independent distributed. $S_n=\sum_i^N X_i$, with $N$ which has a Poisson distribution and is independent from $X_i$. I have to compute the ...
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1answer
22 views

Moment-determinacy in multivariate case

Let $X$ be a random vector with probability density $p$. In the scalar case I have learned that if the characteristic function of $X$ is real analytic, then all moments exist and $p$ is determined ...
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3answers
46 views

What is the set with characteristic function $\chi_A(x) + \chi_B(x)-\chi_A(x)\chi_B(x)$?

Suppose that $A$ and $B$ are subsets of $X$ Find the subset $C$ whose characteristic function is given by: $\chi_C(x)=\chi_A(x) + \chi_B(x)-\chi_A(x)\chi_B(x)$ The answer given is ...
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1answer
119 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
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0answers
105 views

properties of characteristic function

Let $X,Y$ be two independent random variables having the same distribution, centred and with variance 1, $\phi$ is the characteristic function of $X$ and $Y$. If $X+Y$ and $X-Y$ are independent, show ...
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15 views

Characteristic Function of a Spextral Density Function

I am struggling with understanding the link between the Spectral Density Function and the Characteristic Function. In particular, can you find the Characteristic function when only the SPX and ...
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1answer
47 views

Inequality on characteristic functions (probability theory)

Show that for every real characteristic function $\phi(t)$ we have $$1-\phi(2t) \le 4(1-\phi(t))$$ I am not sure where to begin. It seems I am missing some formula or theorem, or is it really that ...
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1answer
56 views

$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$

Hello everybody i need to show following equality $$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$$ Where $(X_i)_i$ are ...
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1answer
33 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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33 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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1answer
78 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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84 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
89 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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32 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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25 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 ...