Julian Newman
  • Member for 10 years, 8 months
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  • London, UK
Definition of random measures
7 votes

By way of introduction: As expressed in some of the comments, I find the "locally compact" assumption possibly a bit too strong. A weaker assumption than having a locally compact second-countable ...

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Can a periodically additively perturbed sinusoidal vector field on the circle have a stable periodic orbit of higher least period?
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4 votes

The claim is true. (As proved here, it is quite unique to sinusoidal vector fields. The same reference also mentions, in its introduction, existing applications of equation (1) with $g$ taking the ...

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Is "conditioning to a sub-$\sigma$-algebra" a measurable operation?
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4 votes

Okay, I think I've worked out that the answer is no, i.e. there exists a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{B}$ such that $\mathbb{E}_\mathcal{G}$ is not universally measurable. (We will ...

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Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra?
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4 votes

I've found the answer - it's NO! The paper I found addressing the question is the following: http://projecteuclid.org/euclid.aop/1175287757 ("0-1 Laws for Regular Conditional Probabilities") ...

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Convergence of Radon Nikodym derivatives
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3 votes

I'm going to assume that your space is locally compact (as well as $\sigma$-compact), so that $X$ is the union of a sequence of compact sets where each lies in the interior of the next. In this case, ...

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Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes
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3 votes

I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order: The answer to Q2 is yes; the structure of the ...

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Can a weaker version of the Hausdorff paradox be proved without AC?
2 votes

Following the suggestion in the first comment below my question (and with the help of the second comment), I can give an example of a scenario that is "even worse" than what I requested, where $A \cup ...

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Taylor's theorem for a composition with $\min:\mathbb R^2\to\mathbb R$ and differentiability Lebesgue almost everywhere
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1 votes

Obviously in the case that $n=1$, we have $s(x,x)=0$ and so if $f'(x) \neq 0$ then $\frac{\partial h}{\partial y_1}$ doesn't exist at $(x,x)$. So I will assume that $n \geq 2$. Answer to Question 1. ...

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Weaker version of the martingale convergence theorem
1 votes

Let $\Omega=\{-1,1\} \times \{-1,1\}$ with the discrete $\sigma$-algebra and uniform measure, let $X(\omega_1,\omega_2)=\omega_2$, and let $$ \mathcal{A}_n \ = \ \left\{ \begin{array}{l l} \sigma(\{\...

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Is it possible for a random nowhere dense closed set to have a positive probability of hitting any given point?
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1 votes

Yes. As in the comments: take $X=\mathbb{S}^1$; and let $\nu$ be the law of the random set constructed by taking a positive-Lebesgue-measure Cantor set $K \subset \mathbb{S}^1$ and rotating $K$ ...

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Is there a generalised version of the Donsker invariance principle for a "sort-of continuous-time-random-walk"?
0 votes

Having read Mateusz Kwaśnicki's answer, I will now write it in my own way: Lemma. Let $S_\infty$ and $T$ be separable metric spaces, and let $(S_j)_{j \in \mathbb{N}}$ be a sequence of Borel subsets ...

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How far can the domain of definition of multiplier operators be extended?
0 votes

I think I can now prove the following (which covers the case requested in the bounty): Theorem. Let $g=P/Q$ for polynomials $P$ and $Q$ where $\mathrm{order}(P) \leq \mathrm{order}(Q)$ and $Q$ has no ...

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Invariant and periodic measures of the random dynamical system on the circle generated by $d\theta_t=dW_t$
0 votes

I have found an answer; it is based on Proposition 3.10 of here. Claim: $\mathbb{P}_W \otimes \lambda$ is the only $\Theta$-invariant probability measure whose projection onto $\Omega$ is $\mathbb{P}...

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Infima of conditional densities after disintegration
0 votes

Well, the question was asked a long time ago, so my answer might not be of much help to the asker any more; but perhaps for the sake of future readers I'll write an answer anyway. Since densities are ...

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