The Annals of Probability

Transforming random elements and shifting random fields

Hermann Thorisson

Full-text: Open access


Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and $X'$ in this space agree on invariant sets if and only if there is a random transformation $\Gamma$ such that $\Gamma X$ has the same distribution as $X'$. Applying this to random fields in d dimensions under site shifts, we show further that these equivalent claims are also equivalent to site-average total variation convergence. This convergence result extends to amenable groups.

Article information

Ann. Probab., Volume 24, Number 4 (1996), 2057-2064.

First available in Project Euclid: 6 January 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B99: None of the above, but in this section 60G60: Random fields

Topological transformation group random field invariant $\omega$-algebra total variation coupling


Thorisson, Hermann. Transforming random elements and shifting random fields. Ann. Probab. 24 (1996), no. 4, 2057--2064. doi:10.1214/aop/1041903217.

Export citation


  • 1 ALDOUS, D. and THORISSON, H. 1993. Shift-coupling. Stochastic Process. Appl. 44 1 14.
  • 2 BERBEE, H. C. P. 1979. Random Walks with Stationary Increments and Renewal Theory. Math. Centre Tract 112. Center for Mathematics and Computer Science, Amsterdam.
  • 3 BOURBAKI, N. 1948. Elements de Mathematiques. Topologie Generale 9. Hermann, Paris. ´ ´ ´ ´
  • 4 BOURBAKI, N. 1951. Elements de Mathematiques. Topologie Generale 3, 4. Hermann, Paris. ´ ´ ´ ´
  • 5 DALEY, D. J. and VERE-JONES, D. 1988. An Introduction to the Theory of Point Processes. Springer, New York.
  • 6 Dy NKIN, E. B. 1978. Sufficient statistics and extreme points. Ann. Probab. 6 705 730.
  • 7 GEORGII, H.-O. 1996. Orbit coupling. Ann. Inst. H. Poincare. To appear. ´
  • 8 GREENLEAF, F. P. 1969. Invariant Means on Topological Groups. Van Nostrand, New York.
  • 9 GREVEN, A. 1987. Coupling of Markov chains and randomized stopping times. I, II. Probab. Theory Related Fields 75 195 212; 431 458.
  • 10 HALMOS, P. R. 1950. Measure Theory. Van Nostrand, New York.
  • 11 MONTGOMERY, D. and ZIPPIN, L. 1955. Topological Transformation Groups. Wiley, New York.
  • 12 TEMPELMAN, A. 1992. Ergodic Theorems for Group Actions. Kluwer, Dordrecht.
  • 13 THORISSON, H. 1994. Shift-coupling in continuous time. Probab. Theory Related Fields 99 477 483.
  • 14 THORISSON, H. 1996. Point-stationarity in d dimensions and Palm theory. Unpublished manuscript.