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
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Known properties of these generalized Cauchy distributions
Consider the following family of normalized probability densities parametrized by the strictly positive integer $k$:
$$
\begin{align}
\begin{aligned}
&f_k(x) = \frac{k}{\pi}\sin\left(\frac{\pi}{2k}...
- 575
1
vote
0
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83
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A Berry-Esseen-type inequality for uniform distribution
Suppose that $X_1, X_2, \cdots ,X_n$ are i.i.d with $X_i \sim U([-\sqrt3,\sqrt3])$, $\Phi(t)=(2\pi)^{-\frac{1}{2}}\int_{-\infty}^{t}e^{-\frac{x^2}{2}}dx$. Show that there exists $C>0$ such that
$$\...
- 11
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0
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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
$$\lim _{T \rightarrow \infty} \...
0
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1
answer
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convolution of step function with box function
Consider the box function: $g_\epsilon=\frac{1}{2\epsilon}\chi_{[-\epsilon,\epsilon]}:\mathbb{R}\rightarrow \mathbb{R}$
for $\epsilon>0$.
For a step function $t$, being a finite linear combination ...
- 113
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0
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Do the ODEs satisfied by characteristic functions have a probabilistic interpretation?
Background
While studying the Poisson distribution, I came across the equation:
$$
\mathbf{E}[\lambda\,g(X)] = \mathbf{E}[X\,g(X-1)],
$$
which holds for a Poisson random variable $X \sim \text{Poisson}...
- 97
2
votes
0
answers
75
views
Ratio of cubic and quadratic form, as elementary symmetric polynomials, is normal?
This is a sequel of Ratio of cubic and quadratic form is approximately normal?
Let be $x_{1},x_{2},..., x_{n}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. ...
1
vote
1
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69
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About characteristic function and their powers
I am studying real analysis. I ended up with the following exercise: suppose that $f \in L^1(0, 1)$, $f \geq 0$ a. e. and suppose that there exists $c \geq 0$ such that
$$\int_0^1 (f(x))^n dx = c, $$
...
- 401
1
vote
1
answer
59
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Proving a multivariate characteristic function formula
I was attempting to prove the characteristic function formula
$$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}) ...
- 35
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2
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Finding characteristic function of $S_\tau$
Suppose $X_n\stackrel{iid}{\sim}U(0,5)$ and let
$$
S_n = X_1 + \ldots+X_n,\quad \tau = \inf\{n\ge 1: X_n \ge 4\}
$$
The problem is to find characteristic functions of $\tau$ and $S_\tau$.
I came up ...
- 153
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0
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65
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Why Laplace transform can compute a characteristic eqution of a PDE
I'm reading some paper of PDE, in which they compute characteristic eqution by Laplace transform.
For example,there is an equation and its boundary condition:
$\partial_t u +\lambda\partial_x u=0$
$u(...
- 41
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votes
0
answers
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Rankine-Hugoniot condition of PDE [duplicate]
The question I'm working on is:
$$u_t + uu_x = 0, t > 0$$
with
$$u(x,0)=f(x)=\begin{cases}
0 & x< 0 \\
1 & 0\leq x\leq1 \\ 0&x>1
\end{cases}$$
I'm trying to find the ...
- 33
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2
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Polya's Condition for characteristic functions
Im reading the proof of the Pólya's condition ( Theorem 4.3.1) in the book "Characteristic Functions , by Eugene Lukacs 2ed".
But i dont understand the following statament in the proof:
...
1
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1
answer
61
views
Expressing characteristic function using covariance
If $X$ is a $n$ dimensional random variable the characteristic function is defined by
$$
\mathbb{E}[e^{i\omega^T x}]= 1 - i\omega^T \mathbb{E}[x] - \frac{1}{2}\omega^T\mathbb{E}[xx^T]\omega + o(\left\...
- 5,731
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If $u(t,x) = \int_0^t \theta(s,x)\;\mathrm{d}s$ and $\theta \geq 0$ is it true that $\chi_{\{\theta > 0\}} = \chi_{\{u > 0\}}$ for $t>0$?
Let $\theta(t,x) \geq 0$ for all $t \in [0,T]$ and $x \in \bar\Omega$ where $\Omega$ is some bounded domain. Define the integral
$$u(t,x) = \int_0^t \theta(s,x)\;\mathrm{d}s$$
Assume that $\theta$ is ...
- 487
0
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1
answer
35
views
Limit of a characteristic function of a semi open interval given by a convergent sequence
Let $\left(x_n\right)_{n\geq1}$ be a sequence of real numbers that converges to $x_0$ and $\chi_{(-\infty, x_n]}$ the characteristic function of the semi open interval $(-\infty,x_n]$. How would we go ...
1
vote
1
answer
189
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Deriving the non-central chi-square PDF from characteristic function
This SE question has a nice answer describing how to derive the PDF of the chi-square distribution by inverting its characteristic function, in an elegant method using contour integrals.
How can we ...
- 175
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Traffic Model: Expansion and Compression waves using Rankine-Hugoniot condition
I am learning PDE and I meet some problems when solving one traffic model which is defined as $ u_t + (1-2u) u_x =0 $ with initial condition $ u(x,0) = f(x) = \begin{cases} 0, x<0\\ 1,x>0\end{...
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Convolution of characteristic functions on double cosets
I'm trying to verify formula (5.1) in Getz's and Hahn's An Introduction to Automorphic Representations, §5.2, p. 138. The goal is to calculate the product of basis elements in the Hecke algebra.
The ...
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Matrix defined in terms of a characteristic function has positive eigenvalues
Let $\mu$ be a Gaussian measure over a Hilbert space $F$ its characteristic function:
$$F(f) = \int e^{i\langle \phi, f\rangle} d\mu(\phi)$$
and define the following matrix:
$$M_{ij} = F(f_i)F(f_j) e^{...
- 7,363
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3
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214
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Characteristic Function and Student's $t$-distribution; a Problem
I'm not sure if it is a mistake in the text or if it is actually meant to be like this, but I'm reading An Intermediate Course in Probability by Gut, and in the chapter on moment generating function (...
- 1,594
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2
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65
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Chf. of a sum of independent r.v. is the product of chfs. Is the converse true?
Let suppose that we have a random variables $X$ and $Y$ whose chfs. can be written as:
$$\forall u \in\mathbb{R},\quad\Phi_X(u) = f(u)\Phi_Y(u).$$
Does it mean that we can write $X=Z+Y$ where $Z$ is a ...
- 518
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Example of characteristic function which is not regularly varying at zero.
Let $X$ a random variable (in $\mathbb{R}$ for simplicity) and $\varphi(t) = \mathbb{E}[e^{i tX}]$ be its characteristic function. If $X$ has a moment of order $2$, then $|\varphi(t)|^2 = 1 - \frac12 \...
- 1,604
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0
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Inversion theorem implication
So, consider inversion theorem in the form $\forall a, b : P(\xi \in \{a, b\}) = 0 \Rightarrow P(\xi \in [a, b]) = \lim_{T\to \infty} \frac{1}{2\pi} \int_{-T}^T \frac{e^{-iat}-e^{-ibt}}{it} \phi_{\xi}(...
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Why is the Fourier transform in probability defined with the opposite sign?
For $f \in L^1(\mathbb{R}^n)$ its Fourier transform is defined as
$$\hat{f}(\xi) = \int_{\mathbb{R}^n} f(x) e^{-i\xi\cdot x} dx$$
up to a choice of normalization. The inverse Fourier transform is ...
- 7,363
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votes
1
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Central limit theorem for two-sided Pareto distribution
I am trying to solve the following problem, which provides an example for a central limit theorem in spite of the fact that the variance is infinite.
Consider the two-sided Pareto distribution with ...
