Questions tagged [characteristic-function]

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58
votes
14answers
8k views

What is the most surprising characterization of the Gaussian (normal) distribution?

A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density: $$ \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ or its characteristic function. As recalled in this ...
40
votes
5answers
13k views

What is the purpose of characteristic functions?

I'm hoping that someone can explain, in layman's terms, what a characteristic function is and how it is used in practice. I've read that it is the Fourier transform of the pdf, so I guess I know what ...
18
votes
1answer
15k views

Link between moment-generating function and characteristic function

I am trying to understand the link between the moment-generating function and characteristic function. The moment-generating function is defined as: $$ M_X(t) = E(\exp(tX)) = 1 + \frac{t E(X)}{1} + \...
13
votes
1answer
381 views

Random variables $X, Z$ such that $Z$ and $\sqrt{X + Z}$ have the same distribution?

I am looking for the distribution of a random variable $Z$ defined as $$Z = \sqrt{X_1+\sqrt{X_2+\sqrt{X_3+\cdots}}} .$$ Here the $X_k$'s are i.i.d. and have same distribution as $X$. 1. Update I ...
11
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2answers
1k views

Central limit theorem proof not using characteristic functions

Is there any proof for the CLT not using characteristic functions, a simpler method? Maybe Tikhomirov or Stein's methods? Something self-contained you can explain to a university student (first year ...
9
votes
2answers
926 views

When to prefer the moment generating function to the characteristic function?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X : \Omega \to \mathbb{R}^n$ be a random vector. Let $P_X = X_* P$ be the distribution of $X$, a Borel measure on $\mathbb{R}^n$. The ...
8
votes
1answer
3k views

How to find a density from a characteristic function?

A distribution has the characteristic function $$\phi(t) = (1-t^2/2)\exp(-t^2/4),\ -\infty \lt t \lt \infty$$ Show that the distribution is absolutely continuous and write the density function of ...
7
votes
1answer
4k views

What is “t” in generating functions?

I am studying generating functions applied to probability (moment generating functions, probability generating functions and characteristic functions). I perfectly see their purposes and usefulnesses, ...
7
votes
2answers
1k views

Why does multiplication in the frequency domain equal convolution in the time domain?

This question came in the context of understanding how to get a distribution of a sum of two iid random variables. I'm working through the top answer to this question Consider the sum of $n$ uniform ...
7
votes
1answer
3k views

characteristic functions and symmetry

If the characteristic function of a random variable is a real-valued function, does this imply that the random variable must be symmetric about zero?
7
votes
1answer
4k views

Characteristic function and Fourier transform

I understand the definition of characteristic functions used in probability theory: For a random Variable $X$ with probability density function $f_X$ the characteristic function is defined as: $$\...
6
votes
2answers
2k views

Characteristic Function of a Compound Poisson Process

The definition of a compound Poisson process and its characteristic function I have are the following: Let $\lambda>0$ and $N\sim\text{Poisson}(\lambda T)$. Also, $\{X_i\}_{i=1}^N$ are i.i.d. ...
5
votes
1answer
136 views

Noncentral chi² with a noncentral chi² noncentrality parameter

Denote by $\chi^2(\nu,\lambda)$ the noncentral chi-square distribution with degrees of freedom $\nu$ and noncentrality parameter $\lambda$. If $\Lambda \sim \chi^2(\nu, 2\theta)$ and $(X \mid \Lambda)...
5
votes
3answers
210 views

Proof of Convergence in Distribution with unbounded moment

I posted the question here, but no one has provided an answer, so I am hoping I could get an answer here. Thanks very much! Prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{...
4
votes
2answers
998 views

Derivation of PMF of Poisson Distribution from its Characteristic Function

I came across a question which asked to obtain the probability function of $X$ (a discrete random variable) with its characteristic function given as follows: $${\phi _X}(t) = {e^{\lambda ({e^{it}} - ...
4
votes
2answers
217 views

Characteristic function problem

Suppose $X_1$ and $X_2$ are independent random variables and suppose also that $X_1$ and $X_1-X_2$ are independent. Show that $$\mathbb{P}_{X_1}[X_1=c]=1$$ for some constant $c$. What I get so ...
4
votes
1answer
174 views

Maximise expectation of exponential given mean and variance

The problem is as follows: Suppose that $X$ is a random variable with $\mathbb{E}X=0$, $\mathbb{D}X = \sigma^2$ and having a finite support: $P(|X|\leq a)=1$. What is the maximum possible value of $\...
4
votes
1answer
2k views

Moment-generating function or characteristic function of univariate skew-t distribution

Is there a moment-generating function or a characteristic function for a univariate skew-t distribution $y\sim ST\left(\xi,\omega^2,\alpha,\nu\right)$ as defined by Azzalini?
4
votes
1answer
166 views

Application of Central Limit Theorem - Uniform Distribution

The following question I found on an old exam: Given $n$ i.i.d. random variables $X_k$, $1 \leq k \leq n$, with uniform distribution on $[-1,1]$, it is easy to compute the characteristic function of ...
4
votes
0answers
69 views

Characteristic function inequality

Random variable $X$ and its characteristic function $\phi_X(t)$ then $$\Pr\left(|X|>\frac2T\right) \leq 2\left(1 - \frac1{2T}\int_{-T}^{T}\phi_X(t)dt\right) $$ I cannot find a way how to ...
3
votes
2answers
81 views

Expected of number of discrete uniform variables whose sum is bigger than k (from characteristic function of discrete Irwin–Hall distribution?)

The problem Imagine we keep uniformly drawing $n$ integers $X_i$ from {0, 1, ..., 9} so that their sum is more than $10$. For instance, one draw would be {1, 0, 2, 5, 3}, hence $n=5$, and repeat this ...
3
votes
1answer
1k views

Characteristic function of the Dirac delta?

What is the characteristic function of the Dirac delta function? Is it $e^{i*0}=1$?
3
votes
1answer
124 views

Prove that $E[g(X)] = \int_{-\infty}^{\infty}G(t)\phi(t) dt$

Let $X$ denote a real-valued random variable with characteristic function $\phi$. Suppose that $g$ is a real-valued function on $\mathbb{R}$ that has the representation $\hspace{25mm}g(x) = \int_{-\...
3
votes
1answer
233 views

Characteristic function of distribution

$$p(x)=e^{-2 |x|}$$ with x in [-inf, +inf]. I've calculated the characteristic function as $E[e^{ikx}]=\frac{1}{ik+2}-\frac{1}{ik-2}=\frac{4}{k^2+4}$. Now i'd like the moments.. so I suppose I should ...
3
votes
1answer
38 views

A question about characteristic functions

A probability 101 question. We know that if two variables $X$ and $Y$ are independent then the characteristic function $\phi_{X+Y}(u)$ can be written as \begin{equation} \phi_{X+Y}(u)=\phi_{X}(u)\phi_{...
3
votes
1answer
1k views

Characteristic function example with Bernoulli and Poisson random variables

Let $\{X_n:n\ge 1\}$ be a sequence of i.i.d. Bernoulli random variables with probability of success $0<p<1$, i.e, $$P\{X_1=1\}=1-P\{X_1=0\}=p.$$ The random variable $Y$ is independent of the ...
3
votes
1answer
304 views

Sampling from characteristic/moment generating function

Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is ...
3
votes
0answers
154 views

Sum of truncated Gammas and degenerate

I have a variable $X$ which I am modelling with a mixture model: $$\begin{aligned} (X|A) &\sim \mathbb{1}_{0 \leq x < w \cdot m} \cdot \frac{\text{Gamma}(\alpha,0,\beta / m)}{k_1} \\ (X|B) &...
3
votes
0answers
478 views

What is the distribution of an affine transformation of a log-normal variable?

Let $X$ be a log-normal variate and $Y = aX + b$ is the affine transformation X. Is $Y$ log-normal? I suspect it is not. Since $X$ is log-normal, its expected value is $$ E[X] = \exp(M + S^2/2) $$ ...
3
votes
0answers
245 views

Characteristic functions can establish stochastic dominance?

In an answer to the question here What is the purpose of characteristic functions? people answered the general question about characteristic functions. One answer mentioned that one can use it to ...
2
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2answers
98 views

Show that $Y_1 X_1 + Y_2 X_2$ $\,{\buildrel d \over =}\,$ $(Y_1^2+Y_2^2)^{1/2}X_1$

I would like verification of my solution to the following problem. QUESTION: Let $X_1, X_2 \,{\buildrel iid \over \sim }\, N(0,1) $ and let $Y_1, Y_2$ be two independent random variables ($X_1, ...
2
votes
2answers
75 views

Expected value of $e^{sP}$ where s is a complex number and $P$ is a Poisson rv

For each positive integer $N$, let $ B_N$ be a binomial $(N,1/3)$ random variable and $P$ be a Poisson(5) random variable. I am trying to understand the statistics of $B_P$. Could someone please hint ...
2
votes
1answer
297 views

How to compute bernoulli distribution PDF from CF

The characteristic function for a Bernoulli distribution is $$\phi(t) = (q+pe^{it}) \text{ where } p+q=1$$ I also know that the relationship between $\phi(t)$ and the pdf $f(k)$ is the Fourier ...
2
votes
1answer
91 views

How would you explain Characteristic Function in layman's terms?

What is a Characteristic Function? Why is it needed? Can you explain it in layman's terms and along with a simple & easy example? Please, limit using formal math notations as far as possible.
2
votes
1answer
80 views

Finding the characteristic function of $Y \sim U(-1,1)$

I know that $\phi_Y(t) = E(e^{itY})=E(\cos(tY))+iE(\sin(tY))$ After integration I have found that $E(\cos(tY))= \frac{\sin(t)}{t}$ and $E(\sin(tY))=0$. So is the characteristic function just $\frac{\...
2
votes
1answer
384 views

Characteristic Function proof

Let $\phi_1,\ldots,\phi_n$ denote characteristic functions for distributions on the real line. Let $a_1,\ldots,a_n$ denote nonnegative constants such that $a_1+\ldots+a_n = 1$. Show that $$\hspace{...
2
votes
1answer
554 views

How to compute the characteristic function of two random variables with different distributions?

What are the steps to obtain the following result: Given that X have $\Gamma(1,s)$ distribution; and that X=x, and Y have the Poisson distribution with parameter x. Then the characteristic function ...
2
votes
1answer
409 views

Deconvolution with fourier transform or characteristic function?

Let us consider the following model: $$Y_j = X_j + \epsilon_j \hspace{15pt} j=1, ..., n$$ Where $Y_j$ is a noisy signal, $\epsilon_j$ is the noise which is independend from the signal $X_j$. We have ...
2
votes
0answers
23 views

finding process corresponding to laplace transform

I have a positive stochastic process $X(t)$ with Laplace transforms $$ \mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b} $$ One can clearly see that the ...
2
votes
0answers
37 views

Good book on characteristic functions that includes the CF-proof of the CLT

The title basically says it all. I would like to learn about CF in order to understand the proof of the CLT that makes use of CF. Ideally I would like to read a book that does not only give proves of ...
2
votes
0answers
104 views

How can you determine if a function is heavy tailed from its characteristic function?

The question is given as follow: Let $N$ have a Poisson distribution with mean $\lambda$. $X_i$ is Cauchy distribution with mode 0 and and scaling parameter $1$. Find the characteristic function ...
2
votes
1answer
434 views

Get Characteristic Function from MGF? [duplicate]

I've been calculating characteristic functions and MGF's and was wondering whether we can always get the characteristic function simply by substituting $it$ instead of $t$ in the resulting equation. ...
1
vote
1answer
48 views

Characteristic function issue

As mentioned in a previous post, I've been trying to work through ALL of the problems in Jacod and Protter's Probability Essentials. The following problem has been giving me issues: Let $Z \sim N(0,...
1
vote
2answers
245 views

Dependent / Independent random variables with identical Cumulative distribution function

I'm stuck with an assignment, hope you guys can help. Question: Show, that there exist random variables $X,Y,X',Y'$ on a Probability Space $(\Omega, \mathscr{F},P)$, so that $X$ and $Y$ are not ...
1
vote
1answer
50 views

What is the characteristic function of a rectified Normal distribution?

Rectified Normal distribution is a hybrid distribution with the following pdf: $f(x;\mu ,\sigma ^{2})=\Phi (-{\frac {\mu }{\sigma }})\delta (x)+{\frac {1}{{\sqrt {2\pi \sigma ^{2}}}}}\;e^{{-{\frac ...
1
vote
1answer
65 views

Finding the characteristic function of a simple linear PDF

X is a random variable with a pdf of $ f(x) = \begin{cases} x/2, & 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases}$ I tried finding the characteristic function of this but ended up with ...
1
vote
1answer
116 views

Poisson binomial distribution-like problem

Given n trials, where, on each trial, you have a given probability of either winning or losing a set amount of money (with both the amount of money and the probability changing for each trial)- what ...
1
vote
1answer
2k views

The mgf and cf of Student's t distribution

A student's t distributed rv $X$ has characteristic function but no moment generating function. I wonder if cf(X)=$E[e^{itX}]$, why we cannot take $t=-iu$ to get the mgf $E[e^{uX}]$? (This question ...
1
vote
0answers
34 views

Why do we use exponent in characteristic function?

A student who is attending probability 101, learned about normal distribution and generating functions recently. We are given a "generating function" as follows: $$G(t)=<e^{itx}>=\int_{-\infty}^...
1
vote
0answers
24 views

How can we write the below characteristic function?

Let us assume that $X$ is a random variable and $a$ is a constant. Now suppose $Y=a+bX$, what would the characteristic function of $Y$ would be? Is it? \begin{eqnarray} \mathbb{E}_X\left[\exp(iuY)\...