Tagged Questions
0
votes
1answer
31 views
How do you invert a characteristic function, when integral does not converge?
I need to find the probability density of some distribution with characteristic function given by:
$$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$
I know the formula for inverting a ...
1
vote
0answers
29 views
P.d.f of a discrete fourier transform of binary variables
Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$.
The discrete fourier transform is defined
$b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
0
votes
0answers
63 views
convolution density of the sum of N random variables
In books it is often stated the convolution that the density of the sum of Xi iid random variables is the convolution:
Assuming i goes from 1 to 4.
We can note the proba density of this sum in ...
13
votes
2answers
163 views
Intuitively, why is the Gaussian the Fourier transform of itself?
It's a standard exercise to find the Fourier transform of the Gaussian $e^{-x^2}$ and show that it is equal to itself. Although it is computationally straightforward, this has always somewhat ...
0
votes
0answers
68 views
Fourier transform of $\log(f(x))$
Suppose for a function $f(x)$ becomes $F(k)$ after a Fourier transform, what is the Fourier transform of $\log(f(x))$? I cannot find any related formula in Fourier transform table or list properties. ...
2
votes
0answers
58 views
The variance of a square integrable function
Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is square integrable, symmetric, has infinite support ($\text{supp}(f)= \mathbb{R}\backslash U$, where $U$ is a set of points), and decays at infinity.
...
1
vote
0answers
51 views
Convolutions of Path Integrals of Gaussian Functions
I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
2
votes
0answers
61 views
Calculating a Poisson probability from the chacteristic function?
In a previous homework assignment we were given a function that corresponds to an arbitrary angular
distribution $A_{FB}=(F-B)/(F+B)=(F-B)/N$,
where F = # of events in the forward hemisphere, B = # of ...
2
votes
2answers
112 views
Inequality regarding difference of characteristic functions
We want to show if $X, Y$ are random variables defined on a common probability space, with characteristic functions $f, g$ respectively, then the following inequality is valid:
$$\sup |f(x)-g(x)| \le ...
3
votes
2answers
113 views
Continuous Random Variable with constant moments?
I would like to know if there exists a measure $\rho$ on the positive real line such that its moments $\int_0^{\infty} x^j d\rho(x)$ are equal to a constant (for example equal to one) for all ...
2
votes
1answer
108 views
Using Khinchin's inequality
At the end of page 5 of the Tao's lectures notes, he sets $\psi$ a Schwartz function supported on the unit cube $[0,1]^n$ and choose $f(x)=\sum_{k=1}^N\epsilon_k\psi(x-ke_1)$, where $e_1$ is one of ...
0
votes
1answer
309 views
How to use joint characteristic function to calculate characteristic function for single variables? [duplicate]
Possible Duplicate:
probability question on characteristic function
It is a problem in my practice exam.
Defined on some common probability space,
two random variables $X$, $Y$ have the
...
0
votes
1answer
107 views
Can you Fourier transform probabilities?
If I have a rect function , and I convolute it with it's self, I get a triangle function. If I convolute with a rect function again, I get a bell-curve. I can continue, so long as I know how to ...
1
vote
0answers
81 views
Singular measures - approximate characteristic function
One can decompose a $\sigma$-finite measure $\mu$ on $\mathbb{R}$ in three parts:
$\mu_{ac}$: absolutely continuous
$\mu_{sc}$: singular continuous
$\mu_{pp}$: pure point
A common example for a ...
2
votes
2answers
66 views
Independence of Random Variables (kernel ICA)
In the paper
Bach, F. R., & Jordan, M. I. (2002). Kernel Independent Component
Analysis. Journal of Machine Learning Research, 3(1), 1-48.
doi:10.1162/153244303768966085
I stumpled upon ...
2
votes
2answers
110 views
Continuity of the Characteristic Function of a RV
Defining the Characteristic Function $ \quad \phi(t) := \mathbb{E} \left[ e^{itx} \right] $ for a random variable with distribution function $F(x)$ in order to show it is uniformly continuous I say
...
8
votes
1answer
189 views
Expectation of a Random Subset of the Roots of Unity.
Let $p$ be a prime. If $1_A(x)$ denotes the indicator function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and $$\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$$ denotes the ...
3
votes
1answer
106 views
Fourier transform inequalities on a probability distribution
I am reading a paper and the following came up:
Given a probability density function, $\rho(x)$, such that for $\epsilon > 0$
$$ \int_{-\infty}^{\infty} |\rho(x)|^{1+\epsilon}dx < \infty ...
2
votes
1answer
115 views
Find the probability of certain measurement for a Laplace Operator on a state function
Let $H$ be the operator $ -\frac{d^{2}}{dx^{2}} $ and let its domain be $$\{f\in L^{2}(\mathbb{R},d\lambda)\text{ }:\int_{-\infty}^{\infty}|xF[f(x)]|^{2}dx<\infty\} $$ where $F$ is the Fourier ...
3
votes
2answers
155 views
Question about computing a Fourier transform of an integral transform related to fractional Brownian motion
I am trying to show an integral transform has a fixed point.
Let $H \in (0,1)$ and consider the following integral transform whose kernel is the density of fractional Brownian motion: $$T_H f(x) = ...
2
votes
1answer
551 views
Recovering the probability mass function from the characteristic function of a discrete probability distribution using Mathematica
I would like to recover the probability mass function (pmf) from the characteristic function (CF) of a discrete probability distribution using Mathematica.
Ideally, I'd like to do calculations like ...
