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To fill in some details in Dan's answer: If $S$ is Polish then there does exist a countable $\mathcal{C} \subset C_b(S)$ which determines weak convergence in $\mathcal{P}(S)$, as you desire. It's a …

Geoffrey Grimmett's Probability on Graphs is an excellent introduction to a variety of current active research areas in discrete probability theory, and is probably at about the level you want (you ma …

Let's be careful with quantifiers. I am reading your second statement as "For every $g$ there exists $h$ such that for all $X,Y$ we have $\mathbb{E}[g(X) \mid Y] = h(X,Y)$". It's not true. Let's ju …

First note the following fact: if $f : [0,T] \to \mathbb{R}$ is an integrable function and $\int_0^t f(s)\,ds = 0$ for all $0 \le t \le T$, then $f = 0$ almost everywhere. (It's immediate that $\int_ …

This is an immediate consequence of the conditional Jensen inequality.

Yes. This can be proved using Dynkin's $\pi$-$\lambda$ theorem. The collection $\mathcal{L} := \{ B \in \Sigma : \mu(B) = \nu(B)\}$ is a $\lambda$-system. By Dynkin's theorem, if $\mathcal{L}$ cont …

As an exercise, prove a Fubini theorem for Itó and Lebesgue integrals: for any $F \in L^2([0,u]^2)$ we have $$\int_0^u \int_0^u F(s,t)\;dR_s\;dt = \int_0^u \int_0^u F(s,t)\;dt\;dR_s.$$ (Hint: start wi …

I'm not sure what you mean by "is there a way to sample". But the following fact may be of interest: Proposition. Let $F_1, F_2, \dots$ be any sequence of distributions (possibly infinite), and …

No, certainly not. Let $Y \sim U(0,1)$, so $Y > 0$ a.s. If $\{X^a : a \in \mathbb{N}\}$ are iid $U(0,1)$, then it is easy to see that $\inf_a X^a = 0$ a.s. In fact this will be true for any $Y$ wit …

It is not true in general. Let $\Omega = V \subset [0,1]$ be a set of outer Lebesgue measure 1 and inner measure $c < 1$ (you may take the complement of a familiar Vitali set of inner measure 0), and …

Edit: Previous answer was bogus. Let me change your notation a little to let $x_0$ be the limit of the $x_n$. Yes, this is true. It follows from tightness and the general fact that $h(x_n, \cdot) \ …

The sharp general result in this direction is the classical law of the iterated logarithm (LIL). Suppose, after renormalizing if necessary, that the $x_n$ are iid with zero mean and unit variance. T …

It's not true. For a counterexample, take $\Omega = \{a,b,c\}$ to be a sample space with 3 points, $\mathcal{F} = 2^{\Omega}$, and $P(A) = \frac{1}{3} |A|$ to be the uniform probability measure assig …

Not an answer, but a few (perhaps trivial) ideas. (Please don't upvote this, because this question should stay "unanswered".) It's known that for any abstract Wiener space, the inclusion $H \subset …

No. It's easy to construct a sequence $Y_n$ with $Y_n \to 0$ a.s. but $E Y_n \to +\infty$. (You can even have $E Y_n \equiv +\infty$ if you wish.) Now let $X$ be a fixed Poisson random variable and $ …