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The tree structures associated with a partition of the sample space $\mathcal{X}$ is usually discussed along with a Beta process(or in computer engineers' world they refer it as "stick-breaking proces …

Usually we find a measure space and look into its bornological stuff, not conversely, a helpful reference may be founded in this book: U. H¨ohle and S. E. Rodabaugh, eds., Mathematics of fuzzy sets: l …

(1) If you assumed that the $Y_{n}^{Q},Y_{\infty}^{Q}$ are all convex(thus continuous) functions along $\mathcal{C}$ being convex, then the uniform convergence follows easily from a classical result, …

The trick is to regard the Wiener measure as a random sample function $f(x,t)$ where $x\in (\Omega, \mathscr{F},P)$ and $t\in \mathscr{T}$ is the time index set. Then the whole stochastic process can …

This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer. On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the …

Hint Use the fact that, if $\alpha_{1},\alpha_{2},\cdots$is a sequence of ordinals satisfying $\alpha_{n}<\Omega$, then there exists an ordinal $\alpha$ such that $\alpha<\Omega$ and $\alpha_{n …

I have to admit that this is the first time I have heard about the term "Choquet Theory" although I heard his name in my topology class earlier. After I read the wikipedia page (Choquet theory) I knew …

Peter Michor's book is exactly what you need, and some of his recent work involves more discussion about reparameterization. Michor, Peter W. Manifolds of differentiable mappings. 1980. Some of …

"D. P. Bertsekas and S. E. Shreve [39, p. 133] use the term Dynkin system, while P. Billingsley [43] and E. B. Dynkin [111] himself use the term $\lambda$-system. B. Fristedt and L. Gray [129, …

Rather than measuring the rate of convergence of a sequence, it is more widely accepted that we measure the rate of convergence of a stochastic process that converges to a stationary measure.For a ge …

To take a different view of the question, what you are asking is whether the sample paths of this Wiener process $W_0$ can be "more smooth" in terms of Hölder continuity. The answer is NO. To make @M …

Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\sig …

I figured it out by looking at the last few chapters in [Shiryayev] and some thoughts. The problem can be considered in following way with the aid of a formalized definition of conditional expectation …