# Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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### Do we have the upper bound or the distribution of the following ratio?

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ ...
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### Representation of characteristic function of Levy process

I'm studying levy processes and i just read this theorem. A few pages before this theorem, there is this one where is the measure with characteristic function: Now my question is, just with ...
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### Second moment of stochastic integral wrt Levy Processes

I have a question about the second moment of the integral wrt Levy Processes. Let Z a Levy processe. We know that: And a few page later is written that by differentiation of the characteristic ...
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### Can we prove the following anti-concentration inequality of polynomials of square Gaussian variables?

Following this question Anti-concentration of Gaussian quadratic form. We have the following inequality: Let $X_1,\dots,X_n$ denote i.i.d. standard Gaussian random variables. For every $\epsilon>0$...
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### A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged

I have recently read about about disintegration theorem, i.e., Disintegration theorem Let $X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$...
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### Riemannian metric induced by a stochastic differential equation

Following this paper, a diffusion process in $\mathcal{R}^d$ $$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$ with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...
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### Is there an inverse Lamperti transformation for diffusions?

The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion. For multidimensional processes there are some conditions on the ...
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