Further comments on the representation problem for stationary processes

Laurent, Stéphane

doi:10.1016/j.spl.2009.12.015

Download

Article (Scientific journals)

Further comments on the representation problem for stationary processes

Laurent, Stéphane

2010 • In Statistics and Probability Letters, 80, p. 592-596

Peer Reviewed verified by ORBi

Permalink
https://hdl.handle.net/2268/77758

DOI
10.1016/j.spl.2009.12.015

Files (1)Send to Details Statistics Bibliography Similar publications

Files

Full Text

RepresentationProblems.pdf

Author preprint (146.98 kB)

Download

All documents in ORBi are protected by a user license.

Send to

RIS BibTex APA Chicago Permalink X Linkedin

Details

Keywords :

Standard filtrations; Long-memory processes

Abstract :

[en] We comment on some points about the coding of stochastic processes by sequences of independent random variables. The most interesting question has to do with the standardness property of the filtration generated by the process, in the framework of Vershik's theory of filtrations. Non-standardness indicates the presence of long memory in a purely probabilistic sense. We aim to provide a short, non-technical presentation of Vershik's theory of filtrations.

Disciplines :

Mathematics

Author, co-author :

Laurent, Stéphane ; Université Catholique de Louvain - UCL

Language :

English

Title :

Further comments on the representation problem for stationary processes

Publication date :

2010

Journal title :

Statistics and Probability Letters

ISSN :

0167-7152

Publisher :

Elsevier Science, Amsterdam, Netherlands

Volume :

Pages :

592-596

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1D-4Y0KW9P-4&_user=532038&_coverDate=04%2F15%2F2010&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1557036560&_rerunOrigin=google&_acct=C000026659&_version=1&_urlVersion=0&_userid=532038&md5=1b378adbadac3f0f2cf3027771f400d7&searchtype=a

Funders :

IAP research network

Available on ORBi :

since 26 November 2010

Statistics

Number of views

169 (11 by ULiège)

Number of downloads

307 (3 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

Bibliography

Blum J., and Hanson D. On a problem in hilbert space with applications. Journal of Mathematics and Mechanics 11 3 (1962) 497-501
Bressaud X., Maass A., Martinez S., and San Martin J. Stationary processes whose filtrations are standard. Annals of Probability 34 4 (2006) 1589-1600
Ceillier, G., 2009. Ph.D. Dissertation, Institut Fourier de Grenoble (in preparation)
Dubins L.E., Feldman J., Smorodinsky M., and Tsirelson B. Decreasing sequences of σ-fields and a measure change for Brownian motion. Annals of Probability 24 2 (1996) 882-904
Émery M. Old and new tools in the theory of filtrations. In: Maass A., Martinez S., and San Martin J. (Eds). Dynamics and Randomness (2002), Kluwer Academic Publishers, Massachusetts 125-146
Émery M. De la théorie de Vershik au mouvement brownien, via des idées de Tsirelson. Séminaire Bourbaki 43 2000/2001 (2000) 63-83 Exp. No. 882
Émery M. On certain almost Brownian filtrations. Annales de l'institut Henri Poincaré (B) Probabilités et Statistiques 41 3 (2005) 285-305
Émery M. Recognising whether a filtration is Brownian: A case study. Séminaire de probabilités XLII. Springer Lectures Notes in Math. vol. 1979 (2009) 383-396
Émery M., and Schachermayer W. On Vershik's standardness criterion and Tsirelson's notion of cosiness. Séminaire de probabilités XXXV. Springer Lectures Notes in Math. vol. 1755 (2001) 265-305
Feldman J., and Rudolph D. Standardness of sequences of σ-fields given by certain endomorphisms. Fundamenta Mathematicae 157 (1998) 175-189
Kalikow S.A. T, T- 1 transformation is not loosely Bernoulli. Annals of Mathematics 115 (1982) 393-409
Kallianpur, G., Wiener, N., 1956. Non-linear prediction. Technical Report 1, Office of Naval Research, Unpublished manuscript
Laurent, S., 2004. Filtrations à temps discret négatif. Ph.D. Dissertation, IRMA, Université de Strasbourg
Laurent, S., (2010a). On standardness and I-cosiness. In: Séminaire de Probabilités XLIII (in press)
Laurent, S., (2010b) On Vershikian and I-cosy random variables and filtrations. In: Teoriya Veroyatnostei i ee Primeneniya (in press)
Leuridan, C., (2009). Filtration d'une suite de variables aléatoires gaussiennes. Privately circulated manuscript
Lévy P. Théorie de l'addition des variables aléatoires. 2nd ed. (1954), Gauthier Villars, Paris
Masani P. Wiener's contribution to generalized harmonic analysis. Bulletin of American Mathematical Society 72 (1966) 73-125
Ornstein D.S. Factors of Bernoulli shifts are Bernoulli shifts. Advances in Mathematics 5 (1970) 349-364
Ornstein D.S. An example of a Kolmogorov automorphism that is not a Bernoulli shift. Advances in Mathematics 10 (1973) 49-62
Rokhlin V.A. On the fundamental ideas of measure theory. American Mathematical Society Translation 71 (1952) 1-53
Rosenblatt M. Stationary processes as shifts of functions of independent random variables. Journal of Mathematics and Mechanics 8 5 (1959) 665-681
Rosenblatt M. Stationary Markov chains and independent random variables. Journal of Mathematics and Mechanics 9 6 (1960) 945-949
Rosenblatt M. The representation of a class of two state stationary processes in terms of independent random variables. Journal of Mathematics and Mechanics 12 5 (1963) 721-730
Rosenblatt M. A comment on a conjecture of N. Wiener. Statistics & Probability Letters 79 (2009) 347-348
Smorodinsky M. Processes with no standard extension. Israel Journal of Mathematics 107 (1998) 327-331
Tsirelson B. Triple points: From non-Brownian filtrations to harmonic measures. Geometric And Functional Analysis 7 (1997) 1096-1142
Vershik A.M. Theorem on lacunary isomorphisms of monotonic sequences of partitions. Funktsional'nyi Analiz i Ego Prilozheniya 2 3 (1968) 17-21 English translation, 1968: Functional Analysis and Its Applications 2 (3) 200-203
Vershik A.M. Decreasing sequences of measurable partitions, and their applications. Doklady Akademii Nauk SSSR 193 (1970) 748-751 English translation, 1970: Soviet mathematics Doklady 11, pp. 1007-1011
Vershik, A.M., 1973. Approximation in measure theory (in Russian). Ph.D. Dissertation, Leningrad Univ. (expanded and updated version: Vershik (1994))
Vershik A.M. The theory of decreasing sequences of measurable partitions (in Russian). Algebra i Analiz 6 4 (1994) 1-68 English translation, 1995: St. Petersburg Mathematical Journal 6 (4) pp. 705-761
Vershik A.M. The Kantorovich metric: The initial history and little-known applications (in Russian). Representation theory, dynamical systems. Part XI, Special issue. Zap. Nauchn. Sem. POMI 312 (2004) 69-85 English translation, 2006: J. Math. Sci. (New York) 133 (4) 1410-1417
Wiener N. Nonlinear Problems in Random Theory (1958), MIT Press, Cambridge, MA