Questions tagged [characteristic-function]

Filter by
Sorted by
Tagged with
59
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 ...
13
votes
1answer
388 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 ...
41
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} + \...
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, ...
3
votes
0answers
254 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 ...
3
votes
1answer
339 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
156 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) &...