# Tagged Questions

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### Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary time series(in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary parts of ...
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### Fourier Transform for option pricing

Can Fourier transforms be used to derive the joint probability density function of stochastic interest rates and sotck price Brownian motions of call options under stochastic interest rates? So lets ...
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### Characteristic Function Inversion

I am studying the relationship / bijection between characteristic functions and CDFs. In particular, given a characteristic function $\phi$ it is posible to recover the cumulative density function ...
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### a generalization of normal distribution to the complex case: complex integral over the real line

How to prove $\int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi}$ for any $t\in \mathbb{R}$? I only obtained the case that $t=0$, $\int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}$. Thanks.
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### 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 ...
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### 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 ...
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### 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?
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### Form of the spectral density in Wiener Khinchin theorem?

The Wienerâ€“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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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) = ...