# Tagged Questions

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### M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, ...
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### Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diļ¬using Particles. The picture below shows the idea how permutations and inversion numbers reflect ...
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### Has anyone seen this series?

I come across the following infinite series. $$\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for t>0 and a>0}.$$ In particular, I am interested in the case where $a=1/4$. ...
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### Pros and cons of probability model for permutations

I am studying probability model of random permetuation Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k inversions ($inv(\pi)$). The analytic approach was considered by ...
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### Characterization of a particular integrable function

Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative ...
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### matrix Khintchine inequality

The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then \begin{equation*} \left( ...
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### fourier decomposition of white noise

I found in some lecture notes that Brownian motion is defined by its Fourier series: $$B(t) = \sum_{m \in \mathbb{Z},\; m \neq 0} \frac{a_m}{m} e^{2\pi i \,mt}$$ Then I would get that its ...
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### multivariate integral calculation in closed form

I am looking for a closed form for the below integral but since I don't have the necessary backgrounds I am not able to solve it: i know the final solution is in the form of modified Bessel functions ...
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### semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix. \[ \int ...
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### Laplace transform on the cone of positive-definite matrices

The title says most. Let $P_p$ be the cone of positive-definite $p \times p$ matrices. One can define the Laplace transform of (the distribution of) a random matrix with values in $P_p$ by (for ...
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### Bochner's Theorem and Total Positivity

Bochner's Theorem for LCA groups applied to the case of $G = U(1)$ and $G^{\vee} = \mathbb{Z}$ tells us that through the Fourier transform, probability measures on the circle are in bijection with ...
I want to embark on a project about inverting a Moment-generating function of a probabilitiy distribution. That is given by $$M_X(t) = \text{E} \exp(tX)$$ Since I have ...