Editing Gaussian process (section)

==RKHS structure and Gaussian process==

Let <math>f</math> be a mean-zero Gaussian process <math>\left\{X_t ; t\in T\right\}</math> with a non-negative definite covariance function <math>K</math> and let <math> R</math> be a symmetric and positive semidefinite function. Then, there exists a Gaussian process <math> X</math> which has the covariance <math> R</math>. Moreover, the [[reproducing kernel Hilbert space]] (RKHS) associated to <math> R </math> coincides with the [[Cameron–Martin theorem]] associated space <math> R(H)</math> of <math> X</math>, and all the spaces <math> R(H)</math>, <math> H_X</math>, and <math>\mathcal{H}(K)</math> are isometric.<ref name="Viitasaari2014">{{cite journal|last1=Azmoodeh|first1= Ehsan | last2=Sottinen | first2= Tommi | last3= Viitasaari | first3 =Lauri |last4= Yazigi| first4=  Adil |title=Necessary and sufficient conditions for Hölder continuity of Gaussian processes|journal= Statistics & Probability Letters | volume=94 |year=2014|pages=230–235|doi=10.1016/j.spl.2014.07.030|arxiv=1403.2215 }}</ref> From now on, let <math>\mathcal{H}(R)</math> be a [[reproducing kernel Hilbert space]] with positive definite kernel <math>R</math>.

Driscoll's zero-one law is a result characterizing the sample functions generated by a Gaussian process:
<math display="block">\lim_{n\to\infty} \operatorname{tr}[K_n R_n^{-1}] < \infty,</math>
where <math>K_n</math> and <math>R_n</math> are the covariance matrices of all possible pairs of <math>n</math> points, implies 
<math display="block">\Pr[f \in \mathcal{H}(R)] = 1.</math>

Moreover, 
<math display="block">\lim_{n\to\infty} \operatorname{tr}[K_n R_n^{-1}] = \infty</math>
implies <ref name="Driscoll1973">{{cite journal|last1=Driscoll|first1=Michael F.|title=The reproducing kernel Hilbert space structure of the sample paths of a Gaussian process|journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete | volume=26|issue=4|year=1973|pages=309–316|issn=0044-3719|doi=10.1007/BF00534894|s2cid=123348980|doi-access=free}}</ref>
<math display="block">\Pr[f \in \mathcal{H}(R)] = 0.</math>

This has significant implications when <math>K = R</math>, as
<math display="block">\lim_{n \to \infty} \operatorname{tr}[R_n R_n^{-1}] = \lim_{n\to\infty}\operatorname{tr}[I] = \lim_{n \to \infty} n = \infty.</math>

As such, almost all sample paths of a mean-zero Gaussian process with positive definite kernel <math>K</math> will lie outside of the Hilbert space <math>\mathcal{H}(K)</math>.