## probability – showing the independence of two random centred Gaussian vectors.

Let $$X = (X_1, …, X_d)$$ be a centred Gaussian vector composed of i.i.d random variables. I have two questions. The first one is whether my approaches correct.

1. I want to show that: $$O$$ being and orthogonal $$dxd$$ matrix, $$OX$$ has the same law as $$X$$.

The way I did was by the following: I say that a general Gaussian vector X has the law $$N(mu_X, sum_X)$$. I wanna show that Y = $$OX$$ has the same law as $$X$$ which is equal to $$O^{-1}X$$. this is saying that $$P_Y(y) propto P_X(O^{-1}X)$$

(leaving the normalisation constant away)$$propto exp(-1/2(O^{-1}Y – mu_X)^T sum_X^{-1}(O^{-1}(Y)-mu_X))$$ = $$exp(-1/2(Y – Omu_X)^T O^{-T} sum_X^{-1}O^{-1}(Y-Omu_X))$$ = $$exp(-1/2 (Y-O mu_X)^T (Osum O^T)^{-1}(Y-Omu_X))$$ which has the law $$N_Y(Omu_X , Osum_XO^T)$$. Therefore this has the same law as X. Is this correct??? And if so what’s the last sentence of my argument?

1. I want to show that when $$a=(a_1,…,a_d)$$ and $$b=(b_1,…,b_d)$$ are two orthogonal vectors on $$R^n$$, Then by considering orthogonal matrix $$O$$, whose first twotwo columns coincide with a and b, show that $$sum_{i=1}^d a_i X_i$$ and $$sum_{i=1}^d b_i X_i$$ are independent. How is this done???

## pr.probability – Computating the expectation of a functional applied to a Gaussian Process

First, a definition : a process $$Z$$ over $$mathbb R^n$$ is said to be a Gaussian Process on $$mathbb R^n$$ with mean function $$m(cdot)$$ and covariance function $$k(cdot, cdot)$$ if for any integer $$k$$ and any set of points $${ x_1, …, x_k } subset mathbb R^n$$, the vector $$(Z(x_i))_{1 leq i leq k}$$ is multivariate Gaussian with mean $$(m(x_i))_{1 leq i leq k}$$ and covariance $$(k(x_i, x_j))_{1 leq i, j leq k}$$.

Given $$Z$$, a Gaussian Process that is almost surely exponentially integrable on $$(0, 1)^n$$, and $$U = (U_i)_{1 leq i leq n}$$ a vector of iid random variables distributed uniformly on $$(0, 1)$$, can I say anything about the following quantity?

$$mathbb{E}_Z left(dfrac{1}{mathbb{E}_U left( e^{Z(U)}right)} right)$$

where $$mathbb{E}_Z$$ (resp $$mathbb{E}_U$$) denotes the expectation taken with respect to $$Z$$ (resp. $$U$$)

So far, I studied sufficient conditions on $$m$$ and $$k$$ for $$Z$$ to be exponentially integrable almost surely, and therefore to have $$mathbb{E}_U left( e^{Z(U)}right) < +infty$$.

However, I don’t know where to start for $$mathbb{E}_Z left(dfrac{1}{mathbb{E}_U left( e^{Z(U)}right)} right)$$, do you have any pointers or interesting references I can read ?

## brownian motion – Compute an integral related with gaussian

Can anyone help me to solve this integral, I think we can try polar coordinate and some property of Gaussian density, but I stuck for long time. Also the WolframAlpha cannot compute this integral.
$$int_{0}^{infty} int_{-2^{n / 2}x}^{infty} frac{1}{2 pisqrt{(j-1)}} exp left(-frac{x^{2}}{2(j-1)}-frac{y^{2}}{2}right) d y d x$$
Thanks a lot!

## linear algebra – Limit spectrum of composition of projections onto random Gaussian vectors

Let $$n > p$$, let $$X in mathbb{R}^{n times p}$$ whose columns $$X_1, ldots, X_p$$ are zero-meaned Gaussian,of covariance $$(rho^{|i – j|})_{i, j in (p)}$$ ($$rho in (0, 1)$$).

Are there (asymptotics or not) known results on the eigenvalue distribution of:

$$left(mathrm{Id} – tfrac{1}{||{X_1}||^2} X_1 X_1^top right) ldots left(mathrm{Id} – tfrac{1}{||{X_p}||^2} X_p X_p^top right) enspace ?$$

From a geometrical point of view, this is the matrix of the application which projects sequentially onto the orthogonal of the span of $$X_p$$, then onto that of $$X_{p-1}$$, etc, so all eigenvalues are in the unit disk, 0 is an eigenvalue, 1 also is an eigenvalue since $$n > p$$.
I would expect the spectrum to “move towards” 1 as $$rho$$ increases, but are there any quantitative results on that?

## reference request – (random fields / gaussian process): On rewritting a certain expectation as a kernel function

Let $$v = (v_1,ldots,v_n)$$ and $$(w_{1,1},ldots,w_{1,n},ldots, w_{n,m})$$ be random vectors with iid coordinates, and also $$v$$ is independent of $$w$$, with $$w_{i,j} sim N(0,1/m)$$ and $$v_j sim N(0,1/n)$$, for $$i=1,ldots,n$$, $$j=1,ldots,m$$. Define a random function $$f_{w,v}:mathbb R^m to mathbb R$$ by $$f_{w,v}(x) := sum_{i=1}^nv_iphi(sum_{j=1}^m w_{i,j}x_j)$$, where $$phi(t):=max(t,0)$$. Fix a point $$a, b in mathbb R^m$$.

Question. How to got about computing
$$p_{m,n}(a,b) := mathbb P(f_{w,v}(x)f_{w,v}(a, b) > 0),$$
Or even just $$lim_{n to infty}lim_{m to infty}p_{m,n}(a,b)$$.

Note that $$p_{n,m}(a,b)$$ is simply the probability that the random field $$x mapsto f_{w,v}(x)$$ flips its sign between $$x=a$$ and $$x=b$$.

## Observations

• Conditioned on $$w$$, we compute (see this math.SE post)
$$mathbb P(f_{w,v}(x)f_{w,v}(a, b) > 0) = kappa_0(phi_w(a),phi_w(b))$$
where $$mathbb R^n ni phi_w(x) := (phi(sum_{j=1}^mw_{ij}x_j))_{1 le i le n}$$, and $$kappa_0(z,z’) := 1-(1/pi)arccos(z^Tz’/|z||z’|)) in (0, 1)$$ is the arc-cosine kernel of order $$0$$.
• Thus, we have the identity

$$p_{m,n}(a,b) = mathbb E_w(kappa_0(phi_w(a),phi_w(b))).tag{2}$$

Question. Can the formula (2) be further simplified ? Is it linked to some other kernels ?

## Does additive Gaussian noise preserves the Shannon entropy ordering?

Suppose that $$Z$$ is a Gaussian random variable independent of $$X$$ and $$Y$$. Moreover suppose that $$h(X) geq h(Y)$$, where $$h(cdot)$$ is the differential Shannon entropy.

Does relation $$h(X+Z) geq h(Y+Z)$$ hold in general? or under some conditions? Definitely, if they are Gaussian, it holds.

## pr.probability – Projection onto column space perturbed by Gaussian noise

Suppose we have a matrix $$Xinmathbb{R}^{mtimes n}$$ (with $$m le n$$) with iid standard Gaussian entries, and suppose we have noise matrix $$Winmathbb{R}^{mtimes n}$$ with iid Gaussian entries, but with some small variance $$sigma_W < 1$$. We know that the columns of $$X$$ and $$X+W$$ are linearly independent almost surely, so they form a basis. I am interested in knowing how the noise $$W$$ changes the projection matrix $$X(X^TX)^{-1}X^T$$ of $$X$$. For instance, denoting $$X_W = X+W$$ and $$x_j$$ as the jth column of $$X$$, can we say anything about $$X_W(X_W^TX_W)^{-1}X_W^Tx_j$$? I am especially interested in references that discuss this kind of problem? I know standard matrix perturbation theory results could be applied on this, but I am looking for more systematic and refined analysis.

## pr.probability – Convolution of two Gaussian mixture model

Suppose I have two independent random variables $$X$$, $$Y$$, each modeled by the Gaussian mixture model (GMM). That is,
$$f(x)=sum _{k=1}^K pi _k mathcal{N}left(x|mu _k,sigma _kright)$$
$$g(y)=sum _{i=1}^N lambda _i mathcal{N}left(y|mu _i,sigma _iright)$$
Now, I would like to know the PDF of another variable $$Z$$, where $$Z=X+Y$$.

Is there anyone who can write the explicit PDF of $$Z$$?

## How do you calculate the expected value of \$Eleft{e^{-|X|}right}\$ e.g. for Gaussian X?

If $$X$$ is a random variable, I would like to be able to calculate something like $$Eleft{e^{-|X|}right}$$
How can I do this, e.g., for a normally distributed $$X$$?

## Can Gaussian elimination find a zero vector when \$m < n\$?

Given $$m$$ rows with $$n$$ variables where $$m < n$$, and given that there exists a combination of some of the rows that gives a zero vector.

Is it promised when computing the reduced-row echelon form, with Gaussian elimination to find a zero vector?