Editing Gaussian process (section)

===Stationary case===

For a stationary Gaussian process <math>X=(X_t)_{t\in\R},</math> some conditions on its spectrum are sufficient for sample continuity, but fail to be necessary. A necessary and sufficient condition, sometimes called Dudley–Fernique theorem, involves the function <math>\sigma</math> defined by
<math display="block"> \sigma(h) = \sqrt{ {\mathbb E} \big[ X(t+h) - X(t) \big]^2 } </math>
(the right-hand side does not depend on <math>t</math> due to stationarity). Continuity of <math>X</math> in probability is equivalent to continuity of <math>\sigma</math> at <math>0.</math> When convergence of <math>\sigma(h)</math> to <math>0</math> (as <math>h\to 0</math>) is too slow, sample continuity of <math>X</math> may fail. Convergence of the following integrals matters:
<math display="block"> I(\sigma) = \int_0^1 \frac{ \sigma(h) }{ h \sqrt{ \log(1/h) } } \, dh = \int_0^\infty 2\sigma( e^{-x^2}) \, dx ,</math>
these two integrals being equal according to [[integration by substitution]] <math display="inline"> h = e^{-x^2}, </math> <math display="inline">x = \sqrt{\log(1/h)} .</math> The first integrand need not be bounded as <math>h\to 0+,</math> thus the integral may converge (<math>I(\sigma)<\infty</math>) or diverge (<math>I(\sigma)=\infty</math>). Taking for example <math display="inline">\sigma( e^{-x^2}) = \tfrac{1}{x^a}</math> for large <math>x,</math> that is, <math display="inline">\sigma(h) = (\log(1/h))^{-a/2}</math> for small <math>h,</math> one obtains <math>I(\sigma)<\infty</math> when <math>a>1,</math> and <math>I(\sigma)=\infty</math> when <math>0 < a\le 1.</math>
In these two cases the function <math>\sigma</math> is increasing on <math>[0,\infty),</math> but generally it is not. Moreover, the condition
{{block indent | em = 1.5 | text ={{vanchor|(*)}} &nbsp; there exists <math>\varepsilon > 0</math> such that <math>\sigma</math> is monotone on <math>[0,\varepsilon]</math>}}
does not follow from continuity of <math>\sigma</math> and the evident relations <math>\sigma(h) \ge 0</math> (for all <math>h</math>) and <math>\sigma(0) = 0.</math>

{{math theorem | name = Theorem 1 | math_statement = Let <math>\sigma</math> be continuous and satisfy [[#(*)|(*)]]. Then the condition <math>I(\sigma) < \infty</math> is necessary and sufficient for sample continuity of <math>X.</math>}}

Some history.<ref name=MarcusShepp72 />{{rp|424}}
Sufficiency was announced by [[Xavier Fernique]] in 1964, but the first proof was published by [[Richard M. Dudley]] in 1967.<ref name=Dudley67 />{{rp|Theorem 7.1}}
Necessity was proved by Michael B. Marcus and [[Lawrence Shepp]] in 1970.<ref name=MarcusShepp70>{{cite journal |first1=Michael B. |last1=Marcus |first2=Lawrence A. |last2=Shepp |author-link2=Lawrence Shepp |title=Continuity of Gaussian processes |journal=[[Transactions of the American Mathematical Society]] |volume=151 |issue= 2|pages=377–391 |year=1970 |doi=10.1090/s0002-9947-1970-0264749-1 |jstor=1995502 |doi-access=free }}</ref>{{rp|380}}

There exist sample continuous processes <math>X</math> such that <math>I(\sigma)=\infty;</math> they violate condition [[#(*)|(*)]]. An example found by Marcus and Shepp <ref name=MarcusShepp70 />{{rp|387}} is a random [[Lacunary function#Lacunary trigonometric series|lacunary Fourier series]]
<math display="block"> X_t = \sum_{n=1}^\infty c_n ( \xi_n \cos \lambda_n t + \eta_n \sin \lambda_n t ) ,</math>
where <math>\xi_1,\eta_1,\xi_2,\eta_2,\dots</math> are independent random variables with [[Normal distribution#Standard normal distribution|standard normal distribution]]; frequencies <math>0<\lambda_1<\lambda_2<\dots</math> are a fast growing sequence; and coefficients <math>c_n>0</math> satisfy <math display="inline">\sum_n c_n < \infty.</math> The latter relation implies

<math display="inline">{\mathbb E} \sum_n c_n ( |\xi_n| + |\eta_n| ) = \sum_n c_n {\mathbb E} [ |\xi_n| + |\eta_n| ] = \text{const} \cdot \sum_n c_n < \infty,</math>

whence <math display="inline">\sum_n c_n ( |\xi_n| + |\eta_n| ) < \infty</math> almost surely, which ensures uniform convergence of the Fourier series almost surely, and sample continuity of <math>X.</math>
[[File:Autocorrelation of a random lacunary Fourier series.svg|thumb|411px|Autocorrelation of a random lacunary Fourier series]]
Its autocovariation function
<math display="block"> {\mathbb E}[X_t X_{t+h}] = \sum_{n=1}^\infty c_n^2 \cos \lambda_n h </math>
is nowhere monotone (see the picture), as well as the corresponding function <math>\sigma,</math>
<math display="block"> \sigma(h) = \sqrt{ 2 {\mathbb E}[X_t X_t] - 2 {\mathbb E}[X_t X_{t+h}] } = 2 \sqrt{ \sum_{n=1}^\infty c_n^2 \sin^2 \frac{\lambda_n h}2 } .</math>