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(Philosophically, the following question is of a similar flavour to A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary, but more "advanced".) Let $(\Om …

Background of the question. One of the problems that arises with Wiener-driven (and more general Lévy-driven) models of noisy systems is that, due to the extremely rapid decay of tails of infinitely d …

Let $(X_n)_{n \geq 0}$ be an i.i.d. sequence of $\{0,1\}$-valued random variables $X_n \sim \mathrm{Bernoulli}(\frac{1}{2})$, i.e. a sequence of independent tosses of a fair coin. Does there exist a …

Let $(G,\circ)$ be a Polish group, with identity $e$. Let $\Omega$ be the set of continuous functions $\omega:\mathbb{R} \to G$ such that $\omega(0)=e$. For each $t \in \mathbb{R}$, define the project …

I have both a more general question (concerning stopping times), and then a more specific application (as described in the title). Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be …

Let $\mathbf{\Omega}=(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space satisfying the Usual Conditions. Let $P \colon [0,\infty) \times \mathbb{R} \t …

[I've decided to rewrite the question, to make the essential point clearer.] Let $\,\mathbb{R}^{[0,\infty)}:=\{(x_t)_{t \geq 0} : x_t \in \mathbb{R} \ \, \forall t\}$. We say that a set $Y \subset \m …

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 subset …

Let $(M_n)_{n \geq 1}$ be a uniformly bounded martingale over a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Define the probability measure $\mu$ on $\mathbb{R}^\mathbb{N}$ to be the law of $( …

Anthony Quas has provided an example of a càdlàg function for which the jumps are not summable: As in https://math.stackexchange.com/questions/10257/, for any $S \subset (0,1]$ and $(x_t)_{t \in S} \i …

This question is motivated by the question https://math.stackexchange.com/questions/4644235/ on Math Stack Exchange. First, I need to define a notion of transfinite summability that I have not seen el …

(The following question arises from my Math.SE question https://math.stackexchange.com/questions/3643865.) Let $\rho$ be a probability measure on $\mathbb{R} \times (0,\infty)$, and writing $\ \pi_ …

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 Ha …

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 proo …

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to f …