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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 …

No, it is not true for simple examples such as standard Brownian motion or a sequence of independent random variables. Suppose $W$ is a standard Brownian motion on the interval $[0,T]$. The covarian …

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 …

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 $\delta = 1-(1-\epsilon^2)^2/3$ …

There's already been some good responses, but I think it's worth adding a very simple example showing what can go wrong if you don't use Polish spaces. Consider $\mathbb{R}$ under its usual topology, …

Consider the following continuous Markov process X, starting from position x if x = 0 then Xt = 0 for all times. if x ≠ 0 then X is a standard Brownian motion starting from x. This is not strong M …

Yes, the Bochner integral does agree with the Lebesgue integral of the sample paths of the process. We can prove this in a slightly more general situation than that asked for in the question. For a p …

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 …

Letting $\mu_n$ be the distribution of a randomly chosen root of a random polynomial $f=c_0+c_1X+\cdots+c_nX^n$ in $\mathbb{C}[X]$ for IID random variables $c_i\in\mathbb{C}$, each chosen with some pr …

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 …

I can't speak for Paul Malliavin's influences, but I do know a bit about Hormander's theorem (by no means, an expert), and it is naturally suited to differentiable manifolds involving largely the idea …

We can show that $$ \mathbb{P}\left(\sum_ia_iX_i^2\le\epsilon\sum_ia_i\right)\le\sqrt{e\epsilon} $$ so that the inequality holds with $c=1/2$ and $C=\sqrt{e}$. For $\epsilon\ge1$ the right hand side …

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 …

Letting $\mu_{a,\Sigma}$ be the Gaussian measure with covariance matrix $\Sigma$ and mean $a$. Then (double) the variation distance can be written as $$ \left\lVert\mu_{a_1,\Sigma_1}-\mu_{a_2,\Sigma_2 …

To further elaborate on my comment, it is a theorem that if $X^1,X^2,\ldots,X^n$ are Lévy processes with respect to a common filtration, all starting from zero, then they are independent if and only i …