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A stochastic process is a collection of random variables usually indexed by a totally ordered set.

5 votes
1 answer
217 views

Do there exist (almost surely) $C^{\infty}$-smooth Gaussian random fields?

Let $d \ge 1$. Do there exist Gaussian random fields on $\mathbb R^d$ which are (almost surely) $C^{\infty}$-smooth, but which are not analytic? If so, what are necessary and sufficient conditions o …
Tom LaGatta's user avatar
  • 8,532
6 votes

Regular Conditional Probability given a natural filtration of a stochastic process

Let's assume that we are working with the canonical probability space $\Omega = D(\mathbb R)$ of càdlàg functions, and $\mathbb P$ is the law of the process. I would doubt that there is a satisfactor …
Tom LaGatta's user avatar
  • 8,532
7 votes
Accepted

Gaussian processes, sample paths and associated Hilbert space.

The question of continuity of a Gaussian process is a rich one with a lot of theory. Let $T$ be a compact index set, and suppose that $X_t$ is a mean-zero Gaussian process with covariance function $c …
Tom LaGatta's user avatar
  • 8,532
6 votes

Brownian bridge interpreted as Brownian motion on the circle

Let $X_t$ be a continuous stochastic process on the circle $S^1$. For any reasonable application, you probably want that the distribution of $X_t$ is invariant under rotations of the circle. If you …
Tom LaGatta's user avatar
  • 8,532