数理解析研究所講究録 1264 巻 2002 年 42-44

42
COMPLEMENT TO “WHAT DIFFERENTIATES STATIONARYSTOCHASTIC PROCESSES FROM ERGODIC ONES: ASURVEY”
MICHEL VALADIER
These pages complete my paper entitled {CWhat differentiates stationary stochastic processes from ergodic ones: asurvey” published in Publication series of the Research Institute of Mathematical Sciences (Kyoto University, ed. Toru Maruyama)(2001), 33-52.
Rohlin [Ro] and Mackey [Mckl] are among the most important references Iforgot. Note that the papers on the subject are tremendously numerous. Iselected
some ones excluding those treating differential geometry (flows) and C’-algebras.
Now follow some comments.
1. Iwrote my paper before reading of [Mck1-2]. About prediction of stationary
processes and the identification of the law of an ergodic process, [Mckl] exposes
another way to reconstruct the process law ffom the complete past of one trajectory;
see specially the bottom of page 204 till the end of Section 5and the top of page 223.
Many comments in [Mckl] go in the same direction as my paper.
2. Among recent books, see Kifer [Ki], Kallenberg [Ka] and McCutcheon
for their short proofs. Moreover Kallenberg [Ka, Theorem 9.12] proves an ergodic
decomposition theorem for finite number of measurable transformations , . . :’which commute; he uses amean spatial ergodic theorem he proved before [Ka,Theorem 9.9].
$[\mathrm{M}\mathrm{c}\mathrm{C}]$$T_{1}$$T_{d}$

3. If Ihad now to rewrite my paper, Iwould emphazise the method of Kryloff andBogoliouboff [KB] and then the possibility to going to some abstract measurablespaces.
In [KB], [Ox], [DGS], the space $K$ in which the process takes its values is compact
metrizable and this is the easiest case to handle. But most processes are unbounded-valued ones, so the proofs of these papers do not directly apply. One can consider
as asubspace of the compact :this “respects” the topology and introduces acompact over-space, but the big drawback is that is not closed in R.
This leads to say some words about isomorphisms. Note that in Ergodic Theory
maybe three notions of isomorphisms can be used : ones t0-0ne map between sets,
ones t0-0ne map between subsets of full measure and isomorphism of the quotient-algebras such as(the setand $B$ inare equivalent if $P(A\triangle B)=0$ ).
See Petersen [Pe,] and (only for the two first notions) [DGS, pp.3-5].
Ageneral hypothesis encountered in most papers is: the spacein whichthe process takes its values is aBorel standard (or Lusin) measurable space, that is$\mathbb{R}$$\mathbb{R}$$\overline{\mathbb{R}}$$\mathbb{R}$$\mathrm{c}\mathrm{r}$$\mathcal{F}/P$$A$$\mathcal{F}$$\mathrm{p}\mathrm{p}.15-17$$(K, \mathcal{K})$1 Isomorphism with $[0, 1]$ is an essential tool in [Ro].

See also [Mah, Th 6p.157].

43

M. VALADIER

ameasurable space isomorphic to aBorel subset of aPolish topological space. Anyis either countable, either has the cardinality of R. InBorel standard space(the tribe being that of all,or tothe first case it is isomorphic to 1,subsets). In the second case it is isomorphic to $([0, 1], B([0,1]))$ . As areference see[DeM, Appendice au chapitre III, Th.80 p.249] (from many authors all properties
isof Borel standard spaces are proved in Kuratowski’s book [Ku] . Hence if
Borel standard there exists acompact metrizable topology on $K$ whose Borel tribe
coincide with C.
For example is Borel standard. Adirect way to check that is isomorphic
defined by $\varphi(x)=x$ on
as ameasurable space to is the following: let.
and any bijection from onto
Among all works treating Borel standard spaces Isingle out Chersi [Che] andDynkin [Dy]. Maybe only Chersi succeeded proving narrow convergence of the; surely this is thanks to the notion he used of Daniell integral.
sequence
Dynkin, whose method is summarized in my paper, develops his idea of sufficient
statistic, and covers with aunified approach several other notions: Gibbs states,
this question see Aldous [Aid]),symmetric laws (de Finetti-Hewitt-Savage
superharmonic functions...
The work of Lauritzen [Lau] could have some connections with [Dy] (this author,
in apreliminary work, his thesis, does not quote Dynkin’s paper. Idid not see thebook [Lau] .
$(K, \mathcal{K})$$\{$$\ldots$$n\}$$\mathrm{N}\mathrm{U}\{\infty\}$$(K, \mathcal{K})$$)$$\mathbb{R}$$\mathbb{R}$$\varphi:\mathbb{R}arrow\overline{\mathbb{R}}$$\overline{\mathbb{R}}$$\mathrm{N}$$\mathbb{R}\backslash \mathrm{N}$$\mathrm{N}\mathrm{U}\{-\infty, +\infty\}$$(Q_{n}^{\omega})_{n}$$-\mathrm{o}\mathrm{n}$$)$where is the di4. For applications to homogenization the acting group is[Fa]and Varadarajan
mension of the domain under consideration. In 1962 Farrell[Var1-2] worked simultaneously and independently in the case when is alocallycompact group ( is not necessarily commutative but it should admit acountabledense subgroup); see also [Dy, Remark p.717]. Both use limit theorems about powers of compositions of conditional expectations. Note that the case of flows (that) was already treated by von Neumann and Kryloff-Bogoliouboff and thatisFomin gave in 1950 some results in this line ([Fo] is in Russian).
$d$$\mathbb{R}^{d}$

$G$

$G$

$G=\mathbb{R}$SOME FORMER REFERENCES FOR THE READER CONVENIENCE[Che]

F. Chersi, An ergodic decomposition of invariant measures for discrete semiflows onstandard Borel spaces, Advanced topics in the theory of dynamical systems. Proceedingsof the International Conference held in Trento, June 1-6, 1987 (G. Fusco, M. Iannelli), Notes and Reports in Mathematics in Science and Engineering,&L. Salvadori,vol. 6, 1989, PP. 75-87.
M. Denker, C. Grillenberger&K. Sigmund, Ergodic theory on compact spaces, Lecture
Notes in Math., vol. 527, Springer-Verlag, Berlin, 1976.
. Dynkin, Sufficient statistics and extreme points, Ann. Probability 6(1978), 705$\mathrm{e}\mathrm{d}\mathrm{s}.$

[DGS]

[Dy]

$\mathrm{E}.\mathrm{B}$730.

[KB]

N. Kryloff&N. Bogoliouboff, La thiorie g\’en\’erale de la mesure dans son application \‘aVetude des systemes dynamiques de la mecanique non liniaire, Ann. of Math. 38 (1937),65-113.
[Mah][N]

D. Maharam, Decompositions of measure algebras and spaces, Trans. Amer. Math. Soc.
69 (1950), 142-160.
J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math.
33 (1932), 587-642, 789-791 (Pages 307-365 in ’]).
$[\mathrm{N}$44COMPLEMENT
J. von Neumann, Collected works, vol. , Operators, Ergodic theory and almost periodic
functions in agroup (. Taub, ed.), Pergamon Press, Oxford, 1961.
J.C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116-136.

$[\mathrm{N}’]$$\mathrm{I}\mathrm{I}$$\mathrm{A}.\mathrm{H}$[Ox]

NEW RBFERENCES
Astar points out areference Icould not see myself.
[Ald]. Aldous, Exchangeability and related topics, \’Ecole ’\’et\’e de Probabiliti de SaintFlour XIII 1983 (. Hennequin, ed.), Lecture Notes in Math., vol. 1117, SpringerVerlag, Belin, 1985, pp. 1-198 (SEE TH. 12.10 PAGE 100).
C. Dellacherie&P.A.
Dellacherie&PA. Meyer
Meyer, Probabilitis et Potentiel, Chapitres, Hermann, Paris,
1975 (English edition: Nort}
North-Holland, Amsterdam, 1978).
K. Jacobs, Lectures Notes on Ergodic Theory, Aarhus Universitet, Matematisk Institut,$\mathrm{D}.\mathrm{J}$$\mathrm{d}$$\mathrm{P}.\mathrm{L}$[DeM]$[^{*}\mathrm{J}\mathrm{a}]$$\mathrm{I}-\mathrm{I}\mathrm{V}$$\mathit{0}\mathrm{J}$1963.
O. Kallenberg, Foundations of modern probability, Springer-Verlag, New York, 1997 (SEE

[Ka]

$|$

[KW1]

TH .9.12
TH.9.
12 PAGB 164).
J. Kerstan &A. Wakolbing
Wakolbinger, Operateurs markoviens et Vensemble de leurs lois deprobabilite\’e invariantes,probabilite
invariantesn, C. R. Acad. Sci. Paris Ser. A291 (1980), 163-166.
J. Kerstan &A. Wakolbinger, $E\eta od:c$ decomposition of probability laws, Z. Wahrsch.
Verw. Gebiete 56 (1981), 399-414.
Y. Kifer, Ergodic theory of random transformations, Progress in Probability and Statistics, 10, Birkh\"auser,Birkhiuser, Boston, 1986 (SEE PROp .2.1 PAGE 23 AND APPBNDIX A. 1 PAGES 191$\mathrm{R}$$[\mathrm{K}\mathrm{W}2]$$\mathrm{W}\mathrm{a}\mathrm{k}\mathrm{o}\mathrm{l}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{g}($$39^{1}$[Ki]$\mathfrak{n}$

199).

[Ku]

C. Kuratowski, Topologie. , Espaces metrisables, espaces complets.
ed., MonografieMatematyczne, vol. 20, Warszawa-Wroclaw, 1948.
. Lauritzen, Extremaland systems of sufficient statistics, Lecture Notes in
Extremal forStatistics, vol. 49, Springer-Verlag, Berlin, 1988.
gd:c$ thc02. Mackey, $E\tau
Ergodic
theory and its significance for statistical mechanics and pmbability theory, Adv. in Math. 12 (1974), 178-268 (SEE PAGES 192-193, 202).
. Mackey, Von Neuman
Neumann and the early days of ergodic theory, The legacy of John
von Neumann (J. Glimm, J. ImpagliazzO&I. Singer,), Proceedings of the Summer
Research Institute held at Hofstra University, Hempstead, New York, May 29-June4,1988., Proc. Sympos. Pure Math. vol. 50, Amer. Math. Soc, Providence,, 1990,pp. 25-38.
R. Math., Ergodic theory and differentiate dynamics, Springer-Verlag, Berlin-New York,1987 (translated kom\‘a teoria ergodica (Portuguese), Rio de Janeiro, 1983).
from Introducao
Intmdt,R. McCutcheon, Elemental methods
in ergodic Ramsey theory, Lecture Notes in Math.,vol.. 1722, Springer-Verlag,
Berlin,(SEB THB THREE LINES PARAGRAPH AFTER THE1999
Springer-Verlag,$\mathrm{I}$$2\mathrm{d}$.

$\mathrm{W}\mathrm{a}\mathrm{r}^{\tau}$[’Lau]$\mathrm{S}.\mathrm{L}$$famd_{\dot{l}}.es$$\mathrm{S}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{r}-\backslash$[Mckl]$\mathrm{G}.\mathrm{W}$$\cdot$$b_{\dot{l}}lity$[Mck2]$\mathrm{G}.\mathrm{W}$$\mathrm{e}\mathrm{d}\mathrm{s}.$$\mathrm{H}$$\mathrm{R}\mathrm{I}$$[^{*}\mathrm{M}\mathrm{a}\mathrm{n}]$$[\mathrm{M}\mathrm{c}\mathrm{C}]$$\mathrm{M}\mathrm{a}\overline{\mathrm{n}}\acute{\mathrm{e}}$$f$$\mathrm{v}\mathrm{o}\mathrm{l}$$\mathrm{B}$43.3).
ORBM 4.3.3).

[Pe]

[Ro]
[Sc]

$theo\varpi$ , Cambridge Studies in
K. Petersen, Ergodic theory
Advanced Mathematics, 2., Cambridge University Press, CaIr.
Cambridge-New York, 1983.
. Rohlin, On the filndamt
fundamental ideas of measure theory, Amer. Math. Soc. TVansl. 71(1952), 1-54 (translated ffon.
from Mat. Sb. 67 (1949), 107-150).
K. Schmidt, A probabilisticprobabilistic]proof of ergodic decomposition, Sankhyi Ser. A40 (1978),$\mathrm{V}.\mathrm{A}$

10-18.

[Shl]
[Sh2]
[Zim]

$\mathrm{D}\acute{\mathrm{B}}$H. Shimomura, Ergodic $dec\{$de omposition of quasi-invariant measures, Publ.
. Inst.
Math. Sci. 14 (1978), 359-38
359-381.
H. Shimomura, Remark on th
the ergodic decomposition of measures, Publ.
. Inst. Math.
Sci. 26 (1990), 861-865.
D. Zimmermann, Uniqueness in ergodic decomposition of invariant probabilities, Illinois
J. Math. 36 (1992), 325-344.
${\rm Res}$${\rm Res}$PARTBMBNT$\mathrm{D}\mathrm{B}$, UNIVBRSIT\’E MONTPBLLIBR$\mathrm{M}\mathrm{A}\mathrm{T}\mathrm{H}\acute{\mathrm{B}}\mathrm{M}\mathrm{A}\mathrm{T}\mathrm{I}\mathrm{Q}\mathrm{U}\mathrm{B}\mathrm{s}-\mathrm{C}\mathrm{A}\mathrm{S}\mathrm{B}051$

BATAILLON, 34095 MONTPELLIER CEDEX 5, FRANCB$E$-mail address: valadier@math.univ-m0ntp2.fr$\mathrm{I}\mathrm{I}$, PLACB$\mathrm{E}\mathrm{U}\mathrm{G}\grave{\mathrm{B}}\mathrm{N}\mathrm{B}$