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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8
votes
2answers
154 views

Does “independence” of moments imply independence?

Suppse you have two random variables $X,Y$ and you are given that for any $m,n$ that: $$E(X^n Y^m) = E(X^n)E(Y^m)$$ Does this imply that $X$ and $Y$ are independent? Are there some condtions on how ...
7
votes
1answer
552 views

For symmetric stable distributions, why is $\alpha \le 2$?

I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact. Suppose we are trying to come up with stable distributions. From the definition, ...
7
votes
1answer
324 views

Laplace transform of a random variable

My professor says that the Laplace transform of a nonnegative RV uniquely determines the RV up to distributional equality among all nonnegative RVs. He says one can argue this by appealing to a fact ...
6
votes
1answer
978 views

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
6
votes
1answer
61 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 ...
5
votes
2answers
931 views

Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
5
votes
1answer
247 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 ...
5
votes
1answer
392 views

Determining if something is a characteristic function

Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$ Question: Given a function, say $\psi(t)$, how does ...
5
votes
1answer
208 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 ...
4
votes
3answers
1k views

Is it a characteristic function?

Can anyone explain how I can prove that either $\phi(t) = |\cos (t)|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.
4
votes
1answer
85 views

Is there any short proof of this classical problem?

Let $X,Y$ be two i.i.d. r.v.'s with zero mean and unit variance. If $X+Y$ and $X-Y$ are independent, then $X$ and $Y$ are both standard normal distributed. Is there any short proof for this problem?
4
votes
1answer
1k views

How to get PDF from characteristic function

I would appreciate if anybody could explain to me with a simple example how to find PDF of a random variable from its characteristic function. Thank you.
4
votes
1answer
49 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, ...
4
votes
1answer
317 views

Combinations of i.i.d Inverse Chi-Square RVs and their characteristic functions

I am working on a few self-study problems in probability/measure theory and am stuck on characteristic functions. I have the following problem: Given: $X_1,\ldots,X_n$ are iid inverse chi-square(1) ...
4
votes
1answer
682 views

Characteristic functions based proof problem.

I am trying to show that if $T$ be a closed bounded interval and $E$ a measurable subset of $T$. Let $\epsilon >0$, then there is a step function $h$ on $T$ and a measurable subset $F$ of $T$ for ...
4
votes
0answers
58 views

Does $|f\sin (x)|$ integrable on $\mathbb{R}$ imply that $|f|$ integrable on $\mathbb{R}$?

I guess not. Because we usually require $|f|$ to be integrable on ℝ so that it has the fourier transform. Can anyone give me an counterexample for the statement in the title? I have searched for ...
4
votes
0answers
102 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}$ are i.i.d. $\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows ...
4
votes
0answers
3k views

Characteristic functions of random variables (Poisson, Gamma, etc.)

My self-study in measure and probability theory as finally brought me to the subject of characteristic functions, and I have not handled these in the past with any rigor at all, so all of this is ...
3
votes
1answer
158 views

Find characteristic function of random variable

Let random variables $X, Y, Z$ independent. $X$ with uniform distribution on $[-a,a]$, $Y$ with Poisson distribution with parameter $ν$, $Z$ with Bernoulli distribution with parameter $p$. Find the ...
3
votes
2answers
1k views

A criterion for independence based on Characteristic function

Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be ...
3
votes
1answer
60 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 ...
3
votes
2answers
33 views

How to use Cayley-Hamiltonian theorem in proving upper bound on linear space $W$?

If $W = span(I,A,A^1,A^2, \dots)$. What is the upper bound on dimension of $W$? All matrices are $n \times n$. I know that the dim($W$) $\leq n$, by the Cayley-Hamiltonian theorem. However, I don't ...
3
votes
1answer
266 views

Random variable with characteristic function $\large\frac{\phi(t)+\phi(-t)}{2}$ [duplicate]

Possible Duplicate: Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$ If $\phi(t)$ is the characteristic function of a random variable $X$, then $\Re(\phi(t))$ is ...
3
votes
1answer
55 views

Can anything be learned about a probability distribution *directly* from its characteristic function?

Some preliminaries: I know that one can take the inverse Fourier transform to get back the pdf...that is not what I am after. My question is whether the characteristic function, qua function, tells us ...
3
votes
1answer
289 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 ...
3
votes
2answers
433 views

Characteristic function for positive part of random variables

I need your help in solving the following problem: I have to calculate the characteristic function for the positive side of a random variable - simplest case: let $Y$ be standard normal and $Y^+ =\max ...
3
votes
1answer
132 views

If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?

The random variables take values in $\mathbb{R}^d$. I have tried to prove this using characteristic functions. Let $\hat{\mu}_{X_n},\hat{\mu}_{Y_n},\hat{\mu}_{Z_n}$ be the characteristic functions of ...
3
votes
1answer
85 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 ...
3
votes
1answer
564 views

Primitive recursive functions and characteristic functions. Methods of proof- examples. Illumination.

I am puzzling over a sentence in an example in a textbook, showing that a function $f$, defined by cases, is primitive recursive. Let $E$ be the set of even natural numbers. The function $f$ defined ...
3
votes
0answers
69 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 ...
3
votes
1answer
116 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 ...
3
votes
1answer
76 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} ...
3
votes
0answers
322 views

Why the probability characteristic function is always exist but moment generation function is not always exist?

I know that the characteristic function is always exist and moment generation function is not always exist instantly but don't know exactly mathematically.
3
votes
0answers
101 views

Integral of the Normal Characteristic Function

The characteristic function of the $N$-variate Normal distribution is $$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ...
3
votes
0answers
476 views

Distribution of the sum of iid Beta-Negative-Binomial random variables

I am facing a problem when trying to calculate the distribution of the sum of iid Beta-Negative-Binomial random variables or for that matter if only parameter $r$ is different. To get a hint to how ...
2
votes
2answers
141 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 ...
2
votes
3answers
39 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 ...
2
votes
1answer
292 views

Convergence of characteristic functions to $1$ on a neighborhood of $0$ and weak convergence

Prove the following statement: $ X_n \Rightarrow 0 $ (convergence in distribution) if and only if $ (\exists\; \epsilon>0: |t|<\epsilon) \;\; \phi_n(t) \rightarrow 1 $, where $\phi_n(t)$ is ...
2
votes
2answers
62 views

Evaluate $\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$

I'm trying to evaluate the integral $$\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$$ where $\chi_{[0,1]}(x)=1$ is the characteristic function, i.e. equals $1$ for $x \in ...
2
votes
1answer
128 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) = ...
2
votes
3answers
57 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 ...
2
votes
1answer
102 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) = ...
2
votes
2answers
865 views

Step Function and Simple Functions

Definitions: Simple Function: Any functions that can written in the form:$$s(x)=\sum_{k=1}^na_n\chi_{A_n}(x).$$ Note the finite terms here. It should follow that neither all simple functions are ...
2
votes
2answers
271 views

How do you invert a characteristic function, when integral does not converge?

I need to find the probability density of some distribution with characteristic function given by: $$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$ I know the formula for inverting a ...
2
votes
2answers
5k views

Characteristic function of exponential and geometric distributions

I'm trying to derive the characteristic function for exponential distribution and geometric distribution. Can you guide me on getting them? Here is my solution so far: Exponential Dist ...
2
votes
1answer
46 views

Showing convergence by using Levy's continuity theorem

Let $X_n \sim \mathrm{Po} (n)$. I want to show that $Y_n = \displaystyle \frac{X_n -n}{ \sqrt{n}}$ for $n \rightarrow \infty$ converges in distribution to a random variable that is standard normal ...
2
votes
1answer
77 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, ...
2
votes
1answer
46 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.
2
votes
1answer
210 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
votes
2answers
151 views

characteristic functions

I need to prove that if $\phi(t)$ if a characteristic function then so is $e^{\lambda(\phi(t) -1)}$ for $\lambda$ > 0 My problem is that I'm stuck at proving uniform continuity. Is it sufficient to ...