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It is true that there is a universal constant $c$ such that $\mathbb E[f(Z)]\le c\mathbb E[f(Y)]$. The best (smallest) value for $c$ that I can prove now is $c=2.187...$, but expect that it holds for …

The answer to your amended question is yes. In fact, for any $\epsilon\in[0,1)$ we have $$ \mathbb{P}(\vert S\vert > \epsilon)\ge (1-\epsilon^2)^2/3. $$ So, we can take $\delta = 1-(1-\epsilon^2)^2/3$ …

This question keeps getting bumped up, so (at long last) I'll convert my comments above to an answer. It is true that every continuous martingale $X$ with stationary independent increments is a Browni …

You could alternatively try defining Gaussian measures as $2$-stable distributions. This does remove any reliance on finite dimensional projections, and even removes reference to topology. Let $V$ be …

This isn't a problem I've looked at before, but I've been thinking about it since reading your post, and there does seem to be an interesting limit. The following looks like it should all work out, bu …

The set $Z_{\mathcal{P^\ast}}$ is never compact except in the case where it is a singleton (or empty). This is for the general case with $S=(S^1,S^2,\ldots,S^d)$ being an $\mathbb{R}^d$-valued semimar …

No. Suppose that $\Omega$ consists of four points arranged in a square, where adjacent points have distance 1 between them and opposite points have distance 2. Specifically, if the points are labeled …

Yes, it is possible to extend the state space with respect to which $Y$ is a Feller process. Then, $X$ will be a dense open subset of the extension $\hat X$. Furthermore, for any initial distribution …

As has been mentioned in previous answers, the moments do not uniquely determine the distributions unless certain conditions are satisfied, such as bounded distributions. One thing you can say, is tha …

No, it is not true that a process W satisfying the properties (1), (3) and (4) has to be a Brownian motion. We can construct a counter-example as follows. This construction is rather contrived, and I …

As I mentioned in my comment - what you are suggesting implies that the probability of being at (3,4) is the same as being at (5,0) for all large $n$. That seems unlikely, and would guess that $C_n=5$ …

As you suggest in the question, there is no such thing as a uniform measure on the unit sphere of infinite dimensional Banach spaces, such as $L^2\equiv L^2([0,1],\lambda)$ (λ=Lebesgue measure). Inst …

This can answered without any complicated maths. It can be related to the following: Imagine you have $N$ marked cards in a pack of $m$ cards and shuffle them randomly. What is the probability that t …

If $\mathbb{E}\left[\lVert[ X\rVert^d\right]$ is finite then the integral in the question is necessarily finite. As mentioned, this holds whenever $\pi$ is radially decreasing. However, in the general …

Yes, in this case it is true that $p$ is a proper martingale! Note that your integrand $\exp\left(-\int_0^tr_u du\right)$ is an adapted, continuous, and decreasing process bounded by 1. So, the follow …