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(1) (Modified from [1]pp.246-247,312-313) First we sample $Z$ from $Unif[0,1)$, since $\mathbb{R}$ is an Archimedes field, for a fixed $n$ we can find such a $k$ that $\frac{k}{2^n}\leq Z< \frac{k+1}{ …

I think a closest inequality in controlling the deviance between two processes is McDiarmid’s Inequality, and it has an extension which also applies to subgaussian random variables Concentration in un …

Question 1 My Question is whether we can write an explicit form of these sets $B_t$ for my setting, for which I can get bounds on entropy numbers? It depends. The thin sets $B_t$ are thinned f …

Actually if all you are concerned with is the smoothness of the sample path, the smoothness of a Gaussian process is completely characterized by its covariance function. The following result provides …

Sorry I probably missed "$(P_t)_{t \ge 0}$ of a Hunt process" when I first composed this answer and that is why confusion arise. I think it is most beneficial if we start from and stick to definitio …

You may want to have a look into Hwang's Lemma which is known as a discrete (sufficient) analogy to Stein's Lemma though it is NOT a characterization. Generally speaking you can yield such a character …

Problem 1 For a general closed form expression of the joint density of order statistics, it is known intractable: A problem that is closely related to the first-passage time problem is that o …

This post is partly inspired by Fourier Coefficients and Hölder Continuity. Typical proofs of the nowhere differentiability of Brownian paths is by contradiction using binary expansion from real anal …

You can briefly read a short introduction to decide whether you are interested in the theoretic side or application side of the research subject. Say Rasmussen's Notes. If you are more interested in …

Actually I do not think it is a good idea to model this with nested Poisson processes. The major problem occurs at the very beginning of your analysis. ...I want to find a mathematical model to de …

There is an excellent issue of Statistical Science that address the James-Stein phenomena from various aspects. https://www.jstor.org/stable/i23208816 Question What does it mean that a James-Stein es …

Again I had to point out my favorite book on diffusion process below. The authors belong to Ito school, so their understanding is quite insightful and consistent with Ito's. The understanding of his s …

Yes, but with the price of additional assumptions on variances $\sigma^2$ of samples $X_t$([1]'s proof can be extended to non-identical variance $\sigma ^2_i$). For example, [1] Theorem 2 strengthens …

To restate your problem in technical terms: What is the approximation of the distribution of order statistics (in OP we can focus on maximum statistics only) of sample from a Dirichlet process of size …

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 …