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Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions. Is there a direct ch …

By Donsker's theorem, this should converge to a Brownian motion in the scaling limit. This means that the shapes Robby McKilliam plotted will converge to a circle (when properly scaled), since the di …

I'm not sure, but I'll bet you can find the answer in the recent book Markov Chains and Mixing Times by Peres, Levin and Wilmer.

Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\s …

Let $M$ be a smooth $n$-dimensional manifold, and let $FM = GL(M)$ indicate its tangent frame bundle. Let $G$ be a fixed linear subgroup of $GL(n)$, and consider the space $\mathcal S$ of all $G$-str …

Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random v …

In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically infi …

The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In applications …

I understand why the set of natural numbers $\mathbb N = \{ 0, 1, 2, \cdots \}$ is equipped with a total order. Indeed, every monoid has a pre-order, where $$n' \succeq n \quad \mathrm{if~and~only~if} …

Let $Q$ be a smooth manifold, and let $G$ be a Lie group which acts smoothly on $Q$ on the left. Question 1: does the group $G$ act naturally on the tangent bundle $TQ \to Q$? My motivation here …

Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where: $G$ is a Lie group, and $H$ …

Ok I think I've got it, following the partition argument indicated by Michael Greinecker in the comments above. Thank you Michael! Please let me know if you spot any errors. Lemma. Let $G$ be a locall …

Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm. I would like to write $\mathbb Z^d = G / H$, where $G$ is the symm …