1
vote
3answers
158 views

Fourier transform of random binary vector

Consider a uniformly chosen random binary vector $V$ with $n$ elements. That is we say $V_i = 0$ with probability $1/2$ and $V_i=1$ with probability $1/2$. What is the probability distribution of the ...
1
vote
0answers
36 views

Periodic probability density functions and Fourier series coefficients

I am reading a book "Statistics of directional data" which deals with probability density functions $f(\theta)$ where $\theta$ represents the angle around the circle and $f(\theta)$ is periodic with ...
0
votes
0answers
19 views

Inverse Fourier Transform of $S_Y(f)$

I have this power spectral density $$ S_Y(f) =\frac{N_0}{4 \pi ^{2} f^{2}}\left [ 1- \cos(2\pi f T) \right ] $$ Can any one help me how to find the Inverse Fourier transform?
1
vote
0answers
207 views

Form of the spectral density in Wiener Khinchin theorem?

The Wiener–Khinchin theorem says the autocorrelation function of a wide sense stationary process can be written as a Stieltjes integral, where the integrator function is called the power spectral ...
0
votes
1answer
111 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
74 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 ...
15
votes
2answers
478 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
166 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. ...
3
votes
2answers
305 views

Characteristic function for positive part of random variables

I need your help in solving the following problem: I have to calculate the characteristic function for the positive side of a random variable - simplest case: let $Y$ be standard normal and $Y^+ =\max ...
2
votes
0answers
70 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
60 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
85 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
149 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
128 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
144 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
486 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
160 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
111 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
86 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 ...
3
votes
2answers
252 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
231 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
144 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
125 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
184 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
687 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 ...