Geodesic
Conditional zero-one laws
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 4, pp. 828-834 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We say that a class of events fulfills a conditional zero-one law if it is a subset of the completion of the conditioning $\sigma$-algebra. In this case the conditional probability of an event of the class is an indicator function. Therefore the conditional probability takes almost surely only the values zero and one; in the unconditional case the indicator functions are almost surely constant. We consider two special zero-one laws. If a sequence of random variables is conditionally independent, then its tail $\sigma$-algebra fulfills a conditional zero-one law; this generalizes Kolmogorov's zero-one law. If the sequence is even conditionally identically distributed, then its permutable $\sigma$-algebra, which contains the tail $\sigma$-algebra, fulfills a conditional zero-one law; this generalizes the zero-one law of Hewitt and Savage.
Keywords: conditional probability, conditional independence, zero-one law. 
@article{TVP_2003_48_4_a12,
     author = {K. Hess},
     title = {Conditional zero-one laws},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {828--834},
     year = {2003},
     volume = {48},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_4_a12/}
}
TY  - JOUR
AU  - K. Hess
TI  - Conditional zero-one laws
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2003
SP  - 828
EP  - 834
VL  - 48
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TVP_2003_48_4_a12/
LA  - en
ID  - TVP_2003_48_4_a12
ER  - 
%0 Journal Article
%A K. Hess
%T Conditional zero-one laws
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2003
%P 828-834
%V 48
%N 4
%U http://geodesic.mathdoc.fr/item/TVP_2003_48_4_a12/
%G en
%F TVP_2003_48_4_a12
K. Hess. Conditional zero-one laws. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 4, pp. 828-834. http://geodesic.mathdoc.fr/item/TVP_2003_48_4_a12/

[1] Aldous D., Pitman J., “On the zero-one law for exchangeable events”, Ann. Probab., 7:4 (1979), 704–723 | DOI | MR | Zbl

[2] Bauer H., Wahrscheinlichkeitstheorie, De Gruyter, Berlin, New York, 1991, 520 pp. | MR | Zbl

[3] Blum J. R., Pathak P. K., “A note on the zero-one law”, Ann. Math. Statist., 43 (1972), 1008–1009 | DOI | MR | Zbl

[4] Chow Y. S., Teicher H., Probability Theory, Springer-Verlag, New York, 1997, 488 pp. | MR

[5] De Finetti B., “La prévision: ses lois logiques, ses sources subjectives”, Ann. Inst. H. Poincaré, 7 (1937), 1–68 | MR

[6] Döhler R., “Zur bedingten Unabhängigkeit zufälliger Ereignisse”, Teoriya veroyatn. i ee primen., 25 (1980), 639–645 | MR | Zbl

[7] Hewitt E., Savage L. J., “Symmetric measures on Cartesian products”, Trans. Amer. Math. Soc., 80 (1955), 470–501 | DOI | MR | Zbl

[8] Horn S., Schach S., “An extension of the Hewitt–Savage Zero-One Law”, Ann. Math. Statist., 41 (1970), 2130–2131 | DOI | MR | Zbl

[9] Hunt G. A., Martingales et processus de Markov, Dunod, Paris, 1966, 145 pp. | MR | Zbl

[10] Letta G., “Sur les théorèmes de Hewitt–Savage et de de Finetti”, Lecture Notes in Math., 1372, 1989, 531–534 | MR

[11] Loev M., Teoriya veroyatnostei, IL, M., 1962, 712 pp. | MR

[12] Port S. C., Theoretical Probability for Applications, Wiley, New York, 1994 | MR

[13] Sendler W., “A note on the proof of the zero-one law of Blum and Pathak”, Ann. Probab., 3 (1975), 1055–1058 | DOI | MR | Zbl