Editing Gaussian process (section)

==Continuity==

For a Gaussian process, [[Continuous stochastic process#Continuity in probability|continuity in probability]] is equivalent to [[Continuous stochastic process#Mean-square continuity|mean-square continuity]]<ref>{{cite book |chapter-url=https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1974.2/ICM1974.2.ocr.pdf |title=Proceedings of the International Congress of Mathematicians |first=R. M. |last=Dudley |author-link=Richard M. Dudley |year=1975 |volume=2 |pages=143–146 |chapter=The Gaussian process and how to approach it}}</ref>{{rp|145}}{{rp|91 "Gaussian processes are discontinuous at fixed points."}}<ref name = "banerjeeGelfandSmoothness">{{cite journal | first1 = Sudipto | last1 = Banerjee | first2 = Alan E. | last2 = Gelfand | title = On smoothness properties of spatial processes | journal = Journal of Multivariate Analysis | year = 2003 | volume = 84 | issue = 1 | pages = 85–100 | doi = 10.1016/S0047-259X(02)00016-7 | url = https://doi.org/10.1016/S0047-259X(02)00016-7}}</ref>
and [[Continuous stochastic process#Continuity with probability one|continuity with probability one]] is equivalent to [[Continuous stochastic process#Sample continuity|sample continuity]].<ref>{{cite book |first=R. M. |last=Dudley |title=Selected Works of R.M. Dudley |chapter=Sample Functions of the Gaussian Process |date=2010 |author-link=Richard M. Dudley |journal=[[Annals of Probability]] |volume=1 |issue=1 |pages=66–103 |doi=10.1007/978-1-4419-5821-1_13 |isbn=978-1-4419-5820-4 |chapter-url=http://projecteuclid.org/euclid.aop/1176997026 }}</ref>
The latter implies, but is not implied by, continuity in probability.
Continuity in probability holds if and only if the [[autocovariance|mean and autocovariance]] are continuous functions. In contrast, sample continuity was challenging even for [[#Stationarity|stationary Gaussian processes]] (as probably noted first by [[Andrey Kolmogorov]]), and more challenging for more general processes.<ref>{{cite book |last=Talagrand  |first=Michel |author-link=Michel Talagrand |title=Upper and lower bounds for stochastic processes: modern methods and classical problems |year=2014 |publisher=Springer, Heidelberg |isbn=978-3-642-54074-5 |url=https://www.springer.com/gp/book/9783642540745|series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics }}</ref>{{rp|Sect. 2.8}}
<ref>{{citation
 | last = Ledoux | first = Michel
 | editor1-last = Dobrushin | editor1-first = Roland
 | editor2-last = Groeneboom | editor2-first = Piet
 | editor3-last = Ledoux | editor3-first = Michel
 | contribution = Isoperimetry and Gaussian analysis
 | doi = 10.1007/BFb0095676
 | location = Berlin
 | mr = 1600888
 | pages = 165–294
 | publisher = Springer
 | series = Lecture Notes in Mathematics
 | title = Lectures on Probability Theory and Statistics: Ecole d'Eté de Probabilités de Saint-Flour XXIV–1994
 | volume = 1648
 | isbn = 978-3-540-62055-6
 | year = 1996}}</ref>{{rp|69,81}}
<ref>{{cite book
 | last = Adler | first = Robert J.
 | title = An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes
 | journal = Lecture Notes-Monograph Series
 | isbn = 0-940600-17-X
 | jstor = 4355563
 | location = Hayward, California
 | mr = 1088478
 | publisher = Institute of Mathematical Statistics
 | volume = 12
 | year = 1990}}</ref>{{rp|80}}
<ref>{{cite journal |title=Review of: Adler 1990 'An introduction to continuity...' |first=Simeon M. |last=Berman |date=1992 |journal=Mathematical Reviews|mr = 1088478}}</ref>
As usual, by a sample continuous process one means a process that admits a sample continuous [[Stochastic process#Modification|modification]].
<ref name=Dudley67>{{cite journal |first=R. M. |last=Dudley |author-link=Richard M. Dudley |title=The sizes of compact subsets of Hilbert space and continuity of Gaussian processes |journal=Journal of Functional Analysis |volume=1 |issue= 3|pages=290–330 |year=1967|doi=10.1016/0022-1236(67)90017-1 |doi-access=free }}</ref>{{rp|292}}
<ref name=MarcusShepp72>{{cite book |chapter-url=https://projecteuclid.org/euclid.bsmsp/1200514231 |title=Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, vol. II: probability theory  |first1=M.B. |last1=Marcus |first2=Lawrence A. |last2=Shepp |author-link2=Lawrence Shepp |year=1972 |publisher=Univ. California, Berkeley |pages=423–441 |chapter=Sample behavior of Gaussian processes|volume=6 |issue=2 }}</ref>{{rp|424}}

===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>