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Let $ S\in\mathcal S_N$ be a $ N\times N$ symmetric matrix over the reals, and introduce the (normalised) gaussian measure $ $ \mathrm d\mu(S):=2^{-\frac 12N}\pi^{-\frac14N(N+1)}\exp\left[-\frac12\operatorname{tr}(S^2)\right]\mathrm dS $ $ where $ $ \mathrm dS:=\prod_{i\le j\le N}\mathrm dS_{ij} $ $ I am interested in the integral $ $ p_n(N):=\int_{\mathcal S_N}\operatorname{tr}(S^{2n})\mathrm d\mu(S) $ $ which is in factRead more

Consider a real-valued Gaussian process $ f$ on some compact domain $ \mathcal{X}$ with mean zero and covariance function $ k(x,x’) \in [0,1]$ (also known as the kernel function). This question concerns a finite collection of points in $ \mathcal{X}$ , namely, a set of sampled points $ \mathbf{x} = [x_1,\dotsc,x_n]^T$ and a query pointRead more

I need to numerically evaluate/approximate high-order moments of high-dimensional Gaussian measures/distributions with given mathematical expectations and covariance matrices. The Gaussian measures have dimension $ d$ with say $ d>1000$ and the moments have order $ k$ with say $ k>>1000$ . Hence the classical Wick-Isserlis theorem/formula looks useless because it would involve an astronomically largeRead more

Draws from Gaussian processes with zero mean and radial basis covariance kernels are smooth almost surely. Is there any work on the distribution of the derivative of a draw from such a GP?Read more

I define an exact gaussian wave function and then use NDSolve to obtain the numerical solution for this wavefunction. Then I add a DirichletCondition to make an impenetrable wall for the numerical gaussian wave function to bounce off. But the numerical solution then gives a distorted wave function well before the wave function even touchesRead more

I have a Gaussian and a Lorentzian function here; gd[v_] = Sqrt[(4*Log[2]/Pi)]*(1/Dw)*Exp[-4*Log[2]*((v – v0)/Dw)^2] gl[v_] = (Lw/2/Pi)/((v – v0)^2 + (Lw/2)^2); I want to get a Voigt function by using the convolution of the above two functions, Voi[v_] = Integrate[gd[v’]*gl[v’ – v], {v’, -Infinity, Infinity}] but it just returns the input, So, how to doRead more

I’m completely new to Mathematica and stuck on how to go about trying to fit a Gaussian function to my data. Pretty clueless and nothing I’ve tried has worked. Can anyone help? I’ve imported the data and plotted it, but not sure what to do next… G’ = Import[“GP S3 F2 G’ LX.txt”, “Table”] ListLinePlot[Derivative1[G],Read more

For a Gaussian process $ (Y_t)_{t\geq 0}$ with Hölder continuous paths, define a new process $ (X_t^\varepsilon)_{t \geq 0}$ via $ $ X_t^\varepsilon = \int_0^t \operatorname{sign}(Y_{s+\varepsilon}-Y_s) \, ds.$ $ I would like to understand the limiting behaviour of $ X_t^\varepsilon$ as $ \varepsilon \to 0$ (a limit in distribution would be enough). In my concreteRead more