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This set is not closed with respect to monotone limits in general: if $\mu_n = \delta_{1/n}$, $\mu = \delta_0$, $f_k = \mathbb{1}_{[1/k,1]}$, $f = \mathbb{1}_{(0,1]}$ then $\int f_k d\mu_n = 0 = \int …

I do not think this is possible at all! Suppose that $n = 3$ and $k = 2$, and let $X_1,X_2,X_3$ be the order statistics of a sample of random variables uniformly distributed on $[0,1]$. Then $\mathbb …

(Too long for a comment.) I doubt there's a closed-form expression, but one can obviously find the distribution recursively. Let us consider a more general scenario, where $a_n$ is chosen uniformly f …

Let $Y = X + 1/X - 2$. Then $Y \geqslant 0$ and $\mathbb{E} Y = 2 + 1 - 2 = 1$. Thus, $$ \mathbb{P}(Y > y) \leqslant \frac{\mathbb{E} Y}{y} = \frac{1}{y} $$ for every $y > 0$. If we choose $y = a + 1/ …

First of all, $X$ being non-lattice is not enough for $X_t$ being absolutely continuous. A simple counter-example is $$X_t = \sum_{n=1}^\infty \frac{N_t^{(n)}}{n!} \, ,$$ where $N_t^{(n)}$ are indepen …

First, a few comments to make sure I understand the question correctly: I believe the correct wording is "stabie under maxima/minima" or "max/min-stable" rather than "closed". (Similarly, I think "sa …

This very statement is a part of Theorem 46.3 in Sato's book Lévy processes and infinitely divisible distributions, in case you need a reference. The proof only uses Borel–Cantelli, as in mike's and F …

You ask what is the minimiser of the functional $$ F(\mu) = \int_{[-a,a]} \int_{[-a,a]} \frac{1}{1+(x-y)^2} \mu(dx) \mu(dy) . $$ As Christian Remling points out, it is unlikely a closed-form expressio …

If I understand correctly, there is a hidden Markov chain, say $(S_t)$, whose state $s = S_t$ describes the rate $\mu_s$ at which signals of the observed counting process $N_t$ arrive. If this is corr …

(I know next to nothing about exotic measure spaces, but here is what I was able to find). This question is very close to Maharam's theorem, which asserts that every complete measure space is "isomor …

This seems to be a standard exercise. Anyway, here is a sketch of the solution. Let $T$ be the hitting time of zero, and $\phi_n(s) = \mathbb{E}(e^{-s T} | X_0 = n)$ be the Laplace transform of $T$. …

The real question is: are $X = (X_t)$ and $Y = (Y_t)$ indeed $K$-valued random variables? This is not so obvious in continuous time, and in fact the answer may depend on the choice of the $\sigma$-fie …

For a rather detailed description of operator-stable laws, you may want to consult Operator-stable laws by Hudson and Mason. Examples of course include all stable laws (where all matrices are multipl …

(A rather trivial remark, but too long for a comment.) Phrase it in a different language: two polynomials $P$ and $Q$ with non-negative coefficients have product equal to $$P(x) Q(x) = 1 + x + x^2 + …

If $\epsilon \geqslant \tfrac{1}{n}$, then $$ \mathbb{P}(|X-Y|<\epsilon) \geqslant \sum_{i=1}^n \mathbb{P}(X, Y \in [\tfrac{i-1}{n}, \tfrac{i}{n}]) = \sum_{i=1}^n (\mathbb{P}(X \in [\tfrac{i-1}{n}, \t …