Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [characteristic-function]

82 questions
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14 answers

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 question it is also the only distribution for which the…
robin girard
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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 it is, but I still don't understand its purpose.…
Nick
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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} + \frac{t^2 E(X^2)}{2!} + \dots + \frac{t^n…
Giuseppe
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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 am looking to find a simple distribution for $X_k$,…
Vincent Granville
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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: $$\phi_X(t) = E(\exp(itX)) = \int_{\mathbf{R}}…
Giuseppe
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11
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2 answers

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 of mathematics or physics) and takes less than one…
skan
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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 characteristic function of $X$ is the…
Artem Mavrin
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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, but I fail to grasp the underlying intuition…
Easymode44
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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 the distribution. Attempt:…
statsguyz
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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 distributions on $[0,1]$, or $Z_n$. Why does the…
Anton
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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. and independent of $N$. And $\{U_i\}_{i=1}^N$ are…
Guilherme Salomé
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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?
bob sherman
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Why can a polynomial of degree $>2$ not be a cumulant generating function?

Why can a polynomial of degree $>2$ not be a cumulant generating function? I read somewhere that this is impossible but can't retrieve the source. The answer by StasK to Higher order generalization of the multivariate normal distribution mentions a…
Arnold Neumaier
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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 $|x|^{-3}$ outside $(-1,1)$, the following is…
NamelessGods
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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 there a way to sample from the distribution other…
lbf_1994
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