# Questions tagged [gaussian]

Gaussian functions / distributions / processes...

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### anti-concentration of multi-linear polynomials in Gaussian variables

A Gaussian variable $X_i\sim {\cal N}(0,1)$ is anti-concentrated in the following sense: for any $\epsilon>0$ we have: $$\mathbf{P}( |X_i| \leq \epsilon ) = O(\epsilon).$$ Hence if we consider a ...
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### L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension

For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator: $M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$ (...
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### Ito formula for fractional BM + drift and supremum bound

Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
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Consider a random process described by the following linear dynamics: $$x_{k+1} = a x_k + n_k,$$ where $|a|<1$ and $n_k$s are i.i.d. standard normal distributed. It is quite easy to prove that $... • 11 1 vote 0 answers 118 views ### Relation satisfied by a Gaussian random variable I want to prove the following relation for$X\sim \mathcal{N}(0,1)$,$x\in \mathbb{R}$and$f(x)=\mathbb{E}[\max(X,x)]$: $$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$ It seems that ... • 171 1 vote 0 answers 72 views ### Bayesian inference of stochastically evolving model parameters I have a question related to self-calibration in radio interferometry, but I will try to phrase it as generic as possible. I have a set of data points,$D = \{ d_{0, t_0}, d_{1, t_0}, ..., d_{M, t_0}, ...
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Let $K(t,s):T^2 \to \mathbb{R}$ be a kernel symmetric and type positive (for every $n$ $\sum^n_{i,j}u_iu_jK(t_i,t_j) \geq 0$ and $(u_1,\dots,u_n) \in \mathbb{R}^n$) where $T$ is any set. Thus, it is ...