Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T03:16:08Z http://mathoverflow.net/feeds/question/35209 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35209/relation-between-regularities-of-the-trajectory-of-a-mean-zero-gaussian-process-a Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator robin girard 2010-08-11T10:29:10Z 2012-05-29T19:58:33Z <p>Let $\xi_t$ be a zero-mean gaussian process on $[0,1]$ with covariance operator $C$. I would like to better understand the relation between the covariance operator and the regularity of the trajectories. </p> <p>I already know that </p> <blockquote> <p><strong>Theorem</strong> (Kolmogorov) If there exists $\alpha>1,C\geq 0$ and $\epsilon>0$ such that $$E[|\xi_t-\xi_s|^{\alpha}] \leq C |t-s|^{1+\epsilon}$$ then there exists a modification of the process that is almost surely Hölder-$\delta$ for $\delta\in ]0,\epsilon/\alpha[$.</p> </blockquote> <p>I would like to know if there are other results in this direction (with other spaces than Holder ?) and especially one that relates directly a norm (spectral norm under stationnarity assumption ?) of the covariance operator and the regularities of the trajectories. </p> <hr> <p>note: if I remember there is a link between the closure of the cameron martin space and the support of the gaussian measure associated to the process... how can I reformulate this to answer my question ? ) </p> http://mathoverflow.net/questions/35209/relation-between-regularities-of-the-trajectory-of-a-mean-zero-gaussian-process-a/35213#35213 Answer by The Bridge for Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator The Bridge 2010-08-11T10:44:38Z 2010-08-11T10:44:38Z <p>Hi Robin </p> <p>You should have a look to the following reference :</p> <p>Adler - Introduction to Continuity, Extrema, and other Topics of Gaussian Processes</p> <p>Best Regards</p> http://mathoverflow.net/questions/35209/relation-between-regularities-of-the-trajectory-of-a-mean-zero-gaussian-process-a/35234#35234 Answer by Tom LaGatta for Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator Tom LaGatta 2010-08-11T14:45:42Z 2010-08-11T15:14:25Z <p>Suppose we already know that the process is continuous with probability one, so that the process takes values in the Banach space $X = C([0,1])$ with distribution $\mathbb P$. The covariance operator $C : X^* \to X$ is then a map from the dual space <code>$X^*$</code> to $X$. The support of the Gaussian measure $\mathbb P$ is then the closure of the image of $X^*$ under $C$: <code>$$\operatorname{supp} \mathbb P = \overline{ CX^* }.$$</code> (This is the main theorem of [<a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.nmj/1118795361" rel="nofollow">Vakhania 1975</a>])</p> <p>Now let's construct the Cameron-Martin space. The operator $C$ defines an inner product on the dual space <code>$X^*$</code> by $$\langle f, g \rangle = f(Cg)$$ for <code>$f, g \in X^*$</code>. The space <code>$X^*$</code> isn't necessarily closed under the topology induced by the inner product, so let $H$ be the Hilbert space completion, and let $\iota : X^* \hookrightarrow H$ be the inclusion map. Define a map $\iota^* : H \hookrightarrow X$ first on the dense subspace $\iota X^* \subseteq H$ by <code>$$\iota^*(\iota f) = Cf,$$</code> and extend continuously to all of $H$. Thus the covariance operator factors as $C = \iota^* \circ \iota$. </p> <p>The Hilbert space $\iota^* H$ is a subspace of $X$, and is called the Cameron-Martin space of the process. Interpreting this in the context of the support of the Gaussian measure $\mathbb P$, we have <code>$$\operatorname{supp} \mathbb P = \overline{\iota^* H},$$</code> so that the closure of the Cameron-Martin space (with respect to the original norm of $X$) is exactly the support of $\mathbb P$. </p> <p>I go into these ideas in more detail in Section 2 of my preprint [<a href="http://arxiv.org/abs/1003.0975" rel="nofollow">LaGatta 2010</a>].</p> <p>Suppose your covariance operator is an integral operator with kernel $c(s,t)$, called the covariance function of the process. That is, if $\mu$ is a Radon measure on $[0,1]$, then <code>$$(C\mu)(s) = \int_0^1 c(s,t) \, d\mu(t).$$</code></p> <p>If we write $c_s(t) = c(s,t)$, then the support of $\mathbb P$ is the closure of the span of the functions $c_s$ in $C([0,1])$. So to answer your question: if you know that the closure <code>$\overline{\operatorname{span}\{c_s\}}$</code> is a space with regularity property $P$, then the process $\xi_t$ satisfies property $P$ with probability one.</p>