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0answers
47 views

Possible values of characteristic functions (Fourier transforms)

Can a characteristic function $\varphi_{X}(u)$ from probability theory (the Fourier transform of a probability measure) ever equal zero for either any value of $x$ or any value of $u$? This has been ...
2
votes
1answer
59 views

How to model a stochastic process, continuous in stepsize, which converges against a simple random walk?

I want to compute the probability distribution for a stochastic process with discrete number of steps, where each real value has a nonvanishing probability to be the next stepsize. And I want to ...
2
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0answers
81 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...
2
votes
1answer
65 views

Sum of Independent Variables = Uniform?

Is there a probability density (or measure), such that the sum of two such independent random variables is distributed uniform? In other words, what is the Inverse Fourier Transform of the ...
2
votes
1answer
41 views

Finding $E(X^r\mid Y)$ of an exponential function

Let $(X,Y)$ denote a two-dimensional random vector with an absolutely continuous distribution with density function $$p(x,y) = \frac{1}{y}\exp(-y), \qquad 0 < x < y < \infty.$$ Find ...
1
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1answer
81 views

Lower bound of Fourier transform

We know the Fourier transform of the Gauss-function: $\displaystyle\int_{\xi\in\mathbb{R}^d}e^{-\pi\, C\,|\xi|^2}e^{2\pi i \xi\cdot X}d\xi=C^{-d/2}e^{-\, \pi\, |X|^2/2}$ for any $C>0$. Then ...
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0answers
63 views

Intregral of exponential of Shannon Entropy Function

Here I am going to ask a similar question as rde asked , that is what is the integral of exponential of entropy function. That is what is the value of $F[H(x)]=\int_{-1}^{+1} e^{ikH(f(x^2))} dx$ ...
2
votes
2answers
76 views

Uniqueness of distribution with moments $M_n$ if $\limsup_{n\to\infty} \frac{1}{n}\sqrt[n]{M_n}$ finite

There's a theorem which states that the moments, i.e. $M_n = \mathbb{E}\left(X^n\right)$, of a distribution uniquely identify the distribution if $$ R := \left(\limsup_{n\to\infty} ...
2
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0answers
171 views

Upper bound on truncation error of a fourier series approximation of a pdf?

Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation: $$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$ ...
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 ...
1
vote
3answers
254 views

Is this function - characteristic function of a random variable?

$\phi(t)= \begin{cases} 1,&\text{if $|x|< 1$;}\\ e^{-(|x|-1)^2} ,&\text{if $|x|\geq1$.} \end{cases} $ Can anyone help?
5
votes
1answer
199 views

Determining if something is a characteristic function

Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$ Question: Given a function, say $\psi(t)$, how does ...
0
votes
1answer
80 views

Expressing the cantor function on $[0,1]$ as a function on $\text{Ternary}([0,1])$

I would to link the simple function and probabilistic approach for the calculation of the Fourier transform of the Cantor function. Let $f:[0,1] \to [0,1]$ be the Cantor function. In the simple ...
2
votes
1answer
65 views

Probabilistic calulation of the Fourier transform of the Cantor function

This is on the same theme as in this post, where the Fourier transform was derived using simple function. Let $f:[0,1] \to [0,1]$ be the Cantor function. Then $f$ is the cumulative distribution of ...
1
vote
2answers
112 views

using central limit theorem

I recently got a tute question which I don't know how to proceed with and I believe that the tutor won't provide solution... The question is Pick a real number randomly (according to the uniform ...
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 ...
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 ...
3
votes
2answers
157 views

Inverse Fourier transform of $\varphi(t) = \exp\left(\int_0 ^ 1 \frac{e^{itx} - 1}{x} \right)$?

I'm looking for the distribution whose Fourier transform is given by $$\varphi(t) = \exp\left(\int_0 ^ 1 \frac{e^{itx} - 1}{x} \right),$$ where as usual $\varphi(t) = \int_{- \infty} ^ \infty e^{itx} ...
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
1answer
215 views

An example of a “pathological” power-spectral density function?

Suppose that we are given a wide-sense stationary random process $X$ with autocorrelation function $R_X(t)$. Power spectral density $S_X(f)$ of $X$ is then given by the Fourier transform of $R_X(t)$, ...
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 ...
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2answers
346 views

Problem concerning continuous probability distribution

How do you prove that the real part of the characteristic function of the continuous probability distribution $f(x)$ is a characteristic function, but the imaginary part is not? The second part is ...
5
votes
1answer
371 views

For symmetric stable distributions, why is $\alpha \le 2$?

I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact. Suppose we are trying to come up with stable distributions. From the definition, ...
2
votes
1answer
115 views

To make Random Variables using characteristic functions

I know the result that says "There is a bijection between distribution functions and characteristic functions". So I was wondering if there are necessary or sufficient conditions on the ...
1
vote
1answer
512 views

How to perform deterministic deconvolution?

Here is my problem: I have a random variable $A$ that is the sum of independent random variables $B$ and $C$, i.e. $A=B+C$. All three random variables are on the real domain. $B$ is a Gaussian with ...
0
votes
1answer
634 views

Characteristic function: fourier transform of probability measure or density?

In many texts charecteristic function is defined as a Fourier transform of probability density (if random variable admits a density function). Also we can define a charecteristic function as Fourier ...