# Tagged Questions

**4**

votes

**1**answer

97 views

### Connection between degree of growth and return probabilities of random walks on Lie groups

Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. ...

**2**

votes

**1**answer

132 views

### Distance measure for noisy $SE(3)$ transforms

I have a transformation $T \in SE(3)$ parameterized by a mean quaternion $q$ with covariance matrix $\Sigma_q \in R^{4\times4}$ and a mean translation $t \in R^3$ with covariance matrix $\Sigma_t \in ...

**5**

votes

**1**answer

181 views

### Haar measure on $O(n)$ reduced to simpler probability space

The background of this question is how a random variable $X$ on the orthogonal group $O(n)$ whose distribution is the normalized Haar measure $\mu$, i.e., $\mu( O(n) ) = 1$, can be realized on a ...

**5**

votes

**4**answers

682 views

### Is a measurable homomorphism on a Lie group smooth?

Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth?
Edit: My original question said "measurable ...

**3**

votes

**3**answers

315 views

### Support of an infinitely divisible measure.

Hello,
if $G$ is a compact Lie group. Let $\mu$ be an infinitely divisible measure on $G$, such that $e$, the neutral element of $G$, is in the support of $\mu$. Is that true that the support of ...

**10**

votes

**1**answer

463 views

### Integration over the orthogonal group

Let $O(N)$ be the orthogonal group, and $a,b,c\in\mathbb N$. The question is:
$$\int_{O(N)}U_{11}^aU_{22}^bU_{33}^cdU=?$$
This is quite a tricky question:
(1) The first thought would go to ...

**3**

votes

**2**answers

308 views

### Sample from a delta-ball in the orthogonal group O(n)

An answer to another question derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region.
For ...

**4**

votes

**0**answers

192 views

### some weird relations among beta random variables

Let $X_i, Y_i, Z_i$ be three iid family of standard normal random variables. Then the following random variable is distributed the same as the first coordinate of a uniform $2$-sphere:
$$ ...

**4**

votes

**1**answer

366 views

### A question on random walks on semisimple groups

Let $G$ be a connected semisimple Lie group without compact factor, $\mu$ be a Borel probability measure on $G$ such that the group generated by $\mathrm{supp}(\mu)$ is Zariski dense in $G$. For ...

**4**

votes

**0**answers

163 views

### average Riemannian distance between Identiity and a random point in SO(n) or SU(n)

I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to ...

**3**

votes

**0**answers

452 views

### Has the Lie group preserving a probability distribution been used in Bayesian statistics?

For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define
$$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$
Here $\operatorname{STO}(n)$ denotes ...