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
125 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
218 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
47 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
619 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
390 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, ...
5
votes
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 ...
5
votes
1answer
243 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 ...
4
votes
3answers
780 views

Is it a characteristic function?

Can anyone explain, how can I prove either $\phi(x) = |\cos t|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.
4
votes
1answer
395 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
225 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
0answers
2k 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 ...
4
votes
0answers
580 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 ...
3
votes
2answers
498 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
212 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
25 views

Characteristic function respect to a product measure.

In my lecture notes of probability theory there is a passage I don't understand: We defined the characteristic function respect to a law of probability of a m-dimensional random variable: ...
3
votes
1answer
229 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
316 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
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 ...
3
votes
1answer
356 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
1answer
74 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
52 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
75 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}\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 that ...
3
votes
0answers
377 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
121 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
2answers
58 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
92 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
1answer
40 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
60 views

Showing $\varphi(t)\neq 0$ when $\varphi$ is a characteristic function of an infinitely divisible distribution

Let $\varphi$ be a characteristic function of an infinitely divisible random variable. Show that $\varphi(t) \neq 0$ for all $t$. Sorry, I have no clue how to do it, because if the exponential is ...
2
votes
2answers
85 views

Uniqueness of distribution with moments $M_n$ if $\limsup_{n\to\infty} \frac{1}{n}\sqrt[n]{M_n}$ finite

There's a theorem which states that the moments, i.e. $M_n = \mathbb{E}\left(X^n\right)$, of a distribution uniquely identify the distribution if $$ R := \left(\limsup_{n\to\infty} ...
2
votes
1answer
141 views

Characteristic function converges pointwise

Is there any sequence of probability distributions such that their characteristic functions converge pointwise, but the sequence of prob. distributions itself does not converge weakly? :S
2
votes
2answers
162 views

Which distribution has the moment-generating function $\frac{\pi t}{\sin \pi t}$

The distribution $F(x) = e^{-e^{-x}}$ has moment-generating function $M_F(t) = \Gamma(1-t)$. From this it follows that the distribution of $X-Y$ for independently $F$-distributed $X,Y$ has the ...
2
votes
1answer
91 views

Missing assumption? (Convergence of random variables and characteristic functions)

Here are two exercises from my Probability book ("Probabilidade: um curso em nível intermediário", by Barry James). (I have translated them from Portuguese to English and modified them a bit.) ...
2
votes
2answers
3k 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
2answers
325 views

Obtaining cumulants using the characteristic function

If a random variable $x$ has a characteristic function $\phi(\omega)$, then the $n^{\mathrm{th}}$ moment of the distribution of $x$, $\mu_n$ can be calculated as: $$\mu_n = ...
2
votes
1answer
104 views

Fourier transform of characteristic function in a sphere

A similar question was asked before for an interval in $\mathbb{R}$. I wonder how to do it for a characteristic function of $\{x\in\mathbb{R}^3:|x|<r\}$ i.e. I want to calculate $$ ...
2
votes
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 ...
2
votes
1answer
104 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} ...
2
votes
1answer
47 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 ...
2
votes
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 $N(0,1)$. ...
2
votes
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 ...
2
votes
3answers
542 views

Solving a recurrence relation with the characteristic polynomial

Consider the sequence $\{a_n\}_{n=0}^\infty$ with $a_0 = 0, a_1 = 1, a_{n+2} = 6a_{n+1} - 9a_{n}$. Using the characteristic polynomial prove $a_{n} = n3^{n-1}$. So I really wasn't sure where to ...
2
votes
1answer
87 views

X,Y are independent RVs with known characteristic functions. Find P(X+Y=2).

X,Y are independent random variables with the following characteristic functions: $ \phi_X(\theta) = \frac{1}{4}e^{i\theta}+\frac{3}{4}e^{i2\theta} \\ \phi_Y(\theta) = ...
2
votes
1answer
90 views

Show that the characteristic that passes through the point $(x,y)$ is given by $y(x)=\frac{1}{2}(x^{-2}-x_{0}^{-2})$

The function $u(x,y)$ satisfies the partial differential eqaution $x^{3}\frac{\partial u}{\partial x}-\frac{\partial u}{\partial y}=0$ Show that the characteristic that passes through the point ...
2
votes
1answer
274 views

measurable function, measurable set, characteristic function, and simple function

Firstly, Definition 1: function f is measurable if we have a sequence of simple function $s_n$ such that $s_n \to f$. Definition 2: a set $A$ is measurable if characteristic function $\chi_A$ is ...
2
votes
1answer
221 views

Applying an inversion technique to Characteristic Functions

I am struggling with this concept (self-study). Could someone show me how to explicitly apply the inversion formula for these examples? I am working through about 15 examples, but these 3 seemed ...
2
votes
1answer
106 views

Characteristic functions (Statistics)

I would greatly appreciate any help with this problem. If $f_1, f_2 , f_3$ are three characteristic functions (in Statistics, e.g $E(\exp(itX)))$ such that $f_1*f_3=f_2*f_3$ for all $t$ and we are ...
2
votes
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 ...
2
votes
0answers
85 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 ...
2
votes
0answers
202 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.
2
votes
1answer
84 views

characteristic function properties

In lecture, we had the following corollary (without proof, unfortunately): If $ A \in (0,2) $ and $X$ is a random variable (real-valued) with the following characteristic: $$ \mathbb P(X > x) = ...