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Results tagged with stochastic-processes
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user 238
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
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 …
- 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 …
- 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 …
- 8,532
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 …
- 8,532