Questions tagged [characteristic-function]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
0answers
30 views

Difference between characteristic function and F-transform

I'm struggling to understand the difference between this two functions. I have this condition: $P_j:=\mathbb{Q}(S_T>K):=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{iuK}f_j(u,x,v)}{iu}]\...
1
vote
1answer
62 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
0answers
56 views

How to pass from {Probability density function, convolution} to {Probability density function, characteristic function}?

In Forsman, W.C. (1986) "Polymers in solution: theoretical considerations and newer methods of characterization", Springer, New York. https://www.springer.com/la/book/9780306421464 page 24, it states:...
0
votes
2answers
123 views

How to modify the mean and variance/dispersion of a given distribution

I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution. Ideally, this adjustment would be implemented through a parametric function ...
2
votes
2answers
69 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 ...
0
votes
0answers
17 views

Proving the characteristic function involving sequence of 2 iid variables [duplicate]

We have an iid sequence with: $X_1,X_2,...$ with mean $\mu$ and variance $\sigma^2$. We have another similar sequence: $$Y_1,Y_2,...$$. We have a sequence $U1,U_2,...$ where, $U_n=1/\sqrt(2(n\sigma^...
5
votes
3answers
172 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|^{...
8
votes
2answers
480 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 ...
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
35 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 ...
3
votes
1answer
176 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
58 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 ...
5
votes
1answer
91 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)...
4
votes
1answer
102 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 ...
0
votes
0answers
82 views

Probability distribution from momenta

I would like to get the probability distribution (either pdf or cdf) for a variable, by knowing the first n-momenta of the distribution. I ask: Is there a standard way to deal with this, and maybe a ...
1
vote
2answers
161 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
0answers
76 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 ...
1
vote
0answers
33 views

Sufficient statistic from characteristic function?

I have a density function $f(;\theta):{\mathbb R}\rightarrow{\mathbb R}_+$, where $\theta\in{\mathbb R}^d$. I know that there is a sufficient statistic $T$ of dimension $d$. If $\varphi_n$ is the ...
1
vote
1answer
99 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 ...
3
votes
1answer
222 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 ...
2
votes
1answer
244 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 ...
0
votes
0answers
98 views

Characteristic function of uniform random variable [duplicate]

I am trying to find out expectation of a function of a uniform random variable. I am given a random variable $x$ that is uniformly distributed over the interval $[0, a]$. I want to find out the ...
0
votes
1answer
45 views

$X \sim$ Poisson$(λ)$. What is the distribution of $X/c$? $(c > 0)$

Application: $X$ is the number of particles in a closed volume. $c$ is a constant that converts from particle count to ($>0$) molar concentration. For various reasons, I want to model $Y = X/c$ ...
0
votes
1answer
75 views

characteristic function of a linear function of a random variable

What are the broad steps required to solve a question like this? Let $Y=aX+b$, where $X\sim\text{Exp}(\lambda)\,,\:\lambda>0$ and find the characteristic function of $Y$.
2
votes
1answer
72 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.
11
votes
2answers
993 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 ...
3
votes
0answers
134 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) &...
5
votes
2answers
1k 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. ...
2
votes
0answers
341 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 $$ E[X] = \exp(M + S^2/2) $$ ...
0
votes
1answer
237 views

Plot the density function of a normal random variable knowing only the characteristic function in R

The characteristic function of a normal random variable with mean $\mu$ and standard deviation $\sigma$ is: $$\begin{alignat*}{1} \hat{\phi}(t) & =e^{i\mu t}e^{-\frac{1}{2}\sigma^{2}t^{2}}\\ &...
1
vote
1answer
40 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,...
3
votes
1answer
938 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 ...
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 ...
5
votes
1answer
2k 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, ...
4
votes
1answer
167 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
2answers
648 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}} - ...
2
votes
1answer
73 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{\...
1
vote
0answers
26 views

Joint cumulants of Zn2 characters

Let $f_{c}:Z_2^n \rightarrow \{-1,1\}$ be the character defined as $f_c(x) = (-1)^{<x,c>}$, where $c,x \in Z_2^n$. It is easy to see that since $f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \...
0
votes
1answer
175 views

Characterizing clusters by separate feature vector scores

Say I have a medium amount of dependent variables in a study. These are scores from questionnaires that have been standardized so all are on a scale from 0 to 1. I have clusters of my patients - ...
2
votes
2answers
93 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, ...
3
votes
0answers
213 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 ...
7
votes
1answer
2k 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?
4
votes
1answer
1k 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?
3
votes
1answer
954 views

Characteristic function of the Dirac delta?

What is the characteristic function of the Dirac delta function? Is it $e^{i*0}=1$?
4
votes
2answers
184 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 ...
2
votes
1answer
298 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. ...
3
votes
1answer
108 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_{-\...
2
votes
1answer
302 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{...
8
votes
1answer
2k 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 ...
1
vote
0answers
614 views

Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...