Questions tagged [characteristic-functions]
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.
1,293 questions
4 votes
2 answers
81 views
Known properties of these generalized Cauchy distributions
Consider the following family of normalized probability densities parametrized by the strictly positive integer $k$: $$ \begin{align} \begin{aligned} &f_k(x) = \frac{k}{\pi}\sin\left(\frac{\pi}{2k}...
- 575
1 vote
0 answers
83 views
A Berry-Esseen-type inequality for uniform distribution
Suppose that $X_1, X_2, \cdots ,X_n$ are i.i.d with $X_i \sim U([-\sqrt3,\sqrt3])$, $\Phi(t)=(2\pi)^{-\frac{1}{2}}\int_{-\infty}^{t}e^{-\frac{x^2}{2}}dx$. Show that there exists $C>0$ such that $$\...
- 11
0 votes
0 answers
35 views
Does the characteristic function of a measure tell us whether it contains an atom? [duplicate]
Let $\mu$ be a probability measure on $\mathbb{R}$ and let $\varphi$ be its characteristic function. Exercise 3.3.3 on Durrett's book about probability tells us that $$\lim _{T \rightarrow \infty} \...
0 votes
1 answer
47 views
convolution of step function with box function
Consider the box function: $g_\epsilon=\frac{1}{2\epsilon}\chi_{[-\epsilon,\epsilon]}:\mathbb{R}\rightarrow \mathbb{R}$ for $\epsilon>0$. For a step function $t$, being a finite linear combination ...
- 113
3 votes
0 answers
84 views
Do the ODEs satisfied by characteristic functions have a probabilistic interpretation?
Background While studying the Poisson distribution, I came across the equation: $$ \mathbf{E}[\lambda\,g(X)] = \mathbf{E}[X\,g(X-1)], $$ which holds for a Poisson random variable $X \sim \text{Poisson}...
- 97
2 votes
0 answers
75 views
Ratio of cubic and quadratic form, as elementary symmetric polynomials, is normal?
This is a sequel of Ratio of cubic and quadratic form is approximately normal? Let be $x_{1},x_{2},..., x_{n}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. ...
1 vote
1 answer
69 views
About characteristic function and their powers
I am studying real analysis. I ended up with the following exercise: suppose that $f \in L^1(0, 1)$, $f \geq 0$ a. e. and suppose that there exists $c \geq 0$ such that $$\int_0^1 (f(x))^n dx = c, $$ ...
- 401
1 vote
1 answer
59 views
Proving a multivariate characteristic function formula
I was attempting to prove the characteristic function formula $$Z_x(\lambda) = \sum_{n=0}^{\infty} \frac{i^n}{n!} \sum_{j_1, ..., j_n} \lambda_{j_1} ... \lambda_{j_n} \langle x(r_{j_1}) ... x(r_{j_n}) ...
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