## Zero-one laws for Gaussian measures on Banach space

HTML articles powered by AMS MathViewer

- by Charles R. Baker PDF
- Trans. Amer. Math. Soc.
**186**(1973), 291-308 Request permission

## Abstract:

Let $\mathcal {B}$ be a real separable Banach space, $\mu$ a Gaussian measure on the Borel $\sigma$-field of $\mathcal {B}$, and ${B_\mu }[\mathcal {B}]$ the completion of the Borel $\sigma$-field under $\mu$. If $G \in {B_\mu }[\mathcal {B}]$ is a subgroup, we show that $\mu (G) = 0$ or 1, a result essentially due to Kallianpur and Jain. Necessary and sufficient conditions are given for $\mu (G) = 1$ for the case where*G*is the range of a bounded linear operator. These results are then applied to obtain a number of 0-1 statements for the sample function properties of a Gaussian stochastic process. The zero-one law is then extended to a class of non-Gaussian measures, and applications are given to some non-Gaussian stochastic processes.

## References

- Charles R. Baker,
*On equivalence of probability measures*, Ann. Probability**1**(1973), 690–698. MR**368126**, DOI 10.1214/aop/1176996895 - R. H. Cameron and Ross E. Graves,
*Additive functionals on a space of continuous functions. I*, Trans. Amer. Math. Soc.**70**(1951), 160–176. MR**40590**, DOI 10.1090/S0002-9947-1951-0040590-8 - R. L. Dobrušin,
*Properties of sample functions of stationary Gaussian processes*, Teor. Verojatnost. i Primenen.**5**(1960), 132–134 (Russian, with English summary). MR**0139208** - J. L. Doob,
*Stochastic processes*, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR**0058896** - R. G. Douglas,
*On majorization, factorization, and range inclusion of operators on Hilbert space*, Proc. Amer. Math. Soc.**17**(1966), 413–415. MR**203464**, DOI 10.1090/S0002-9939-1966-0203464-1 - A. M. Garsia, E. Rodemich, and H. Rumsey Jr.,
*A real variable lemma and the continuity of paths of some Gaussian processes*, Indiana Univ. Math. J.**20**(1970/71), 565–578. MR**267632**, DOI 10.1512/iumj.1970.20.20046 - I. M. Gel′fand and A. M. Yaglom,
*Calculation of the amount of information about a random function contained in another such function*, Amer. Math. Soc. Transl. (2)**12**(1959), 199–246. MR**0113741** - Jaroslav Hájek,
*On linear statistical problems in stochastic processes*, Czechoslovak Math. J.**12(87)**(1962), 404–444 (English, with Russian summary). MR**152090** - Kiyosi Itô and Makiko Nisio,
*On the oscillation functions of Gaussian processes*, Math. Scand.**22**(1968), 209–223 (1969). MR**243597**, DOI 10.7146/math.scand.a-10885 - Naresh C. Jain and G. Kallianpur,
*A note on uniform convergence of stochastic processes*, Ann. Math. Statist.**41**(1970), 1360–1362. MR**272050**, DOI 10.1214/aoms/1177696914 - Naresh C. Jain,
*A zero-one law for Gaussian processes*, Proc. Amer. Math. Soc.**29**(1971), 585–587. MR**278369**, DOI 10.1090/S0002-9939-1971-0278369-2 - G. Kallianpur,
*Zero-one laws for Gaussian processes*, Trans. Amer. Math. Soc.**149**(1970), 199–211. MR**266293**, DOI 10.1090/S0002-9947-1970-0266293-4 - E. J. Kelly, I. S. Reed, and W. L. Root,
*The detection of radar echoes in noise. I, II*, J. Soc. Indust. Appl. Math.**8**(1960), 309–341, 481–507. MR**129501** - J. Kuelbs,
*Gaussian measures on a Banach space*, J. Functional Analysis**5**(1970), 354–367. MR**0260010**, DOI 10.1016/0022-1236(70)90014-5 - H. J. Landau and L. A. Shepp,
*On the supremum of a Gaussian process*, Sankhyā Ser. A**32**(1970), 369–378. MR**286167** - Edith Mourier,
*Eléments aléatoires dans un espace de Banach*, Ann. Inst. H. Poincaré**13**(1953), 161–244 (French). MR**64339** - K. R. Parthasarathy,
*Probability measures on metric spaces*, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR**0226684** - T. S. Pitcher,
*On the sample functions of processes which can be added to a Gaussian process*, Ann. Math. Statist.**34**(1963), 329–333. MR**146886**, DOI 10.1214/aoms/1177704271 - Ju. A. Rozanov,
*Infinite-dimensional Gaussian distributions*, Trudy Mat. Inst. Steklov.**108**(1968), 136 pp. (errata insert) (Russian). MR**0298752** - Nicolas Vakhania,
*Sur une propriété des répartitions normales de probabilités dans les espaces $l_{p}\,(1\leq p<\infty )$ et $H$*, C. R. Acad. Sci. Paris**260**(1965), 1334–1336 (French). MR**174075** - Nicolas Vakhania,
*Sur les répartitions de probabilités dans les espaces de suites numériques*, C. R. Acad. Sci. Paris**260**(1965), 1560–1562 (French). MR**174076**

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**186**(1973), 291-308 - MSC: Primary 60G15; Secondary 28A40
- DOI: https://doi.org/10.1090/S0002-9947-1973-0336796-5
- MathSciNet review: 0336796