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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 …

I would like to know what the asymptotic limiting behavior is for the following random walk on $\mathbb Z^d$. By Donsker's invariance principle, I suspect that its behavior is diffusive, i.e., the dis …

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 $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space? Here is one natural …

Let $X_i$ be a sequence of identically-distributed random variables with finite-range dependence (i.e. there exists $I$ such that if $|i-i'| \ge I$, then $X_i$ and $X_{i'}$ are independent), and a fin …

I think Lyons answers this in Russell Lyons - Strong Laws of Large Numbers for Weakly Correlated Random Variables.

Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta_x$ are the Gaussian measure with zero cova …

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 …

Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be …

In addition to the excellent answers above, I also suggest the nice survey Oxtoby, Ergodic Sets (Zbl 0046.11504, MR47262, DOI: 10.1090/S0002-9904-1952-09580-X). Introduction. Ergodic sets were introd …

Suppose we already know that the process is continuous with probability one, so that the process takes values in the Banach space $X = C([0,1])$ with distribution $\mathbb P$. The covariance operator …

Gil, thanks for bumping this post. I think I've got a new idea for you, but it's not a proof yet. Let $\gamma_n$ be a minimizing geodesic between $(-n,0)$ and $(n,0)$, and let $\gamma^{\pm}_n$ be a …

Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real numbe …

Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian me …

This is a sequel to another question I have asked. The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; someti …