## Electronic Journal of Probability

### Concrete Representation of Martingales

Stephen Montgomery-Smith

#### Abstract

Let $(f_n)$ be a mean zero vector valued martingale sequence. Then there exist vector valued functions $(d_n)$ from $[0,1]^n$ such that $\int_0^1 d_n(x_1,\dots,x_n)\,dx_n = 0$ for almost all $x_1,\dots,x_{n-1}$, and such that the law of $(f_n)$ is the same as the law of $(\sum_{k=1}^n d_k(x_1,\dots,x_k))$. Similar results for tangent sequences and sequences satisfying condition (C.I.) are presented. We also present a weaker version of a result of McConnell that provides a Skorohod like representation for vector valued martingales.

#### Article information

Source
Electron. J. Probab., Volume 3 (1998), paper no. 15, 15 pp.

Dates
Accepted: 2 December 1998
First available in Project Euclid: 29 January 2016

https://projecteuclid.org/euclid.ejp/1454101775

Digital Object Identifier
doi:10.1214/EJP.v3-37

Mathematical Reviews number (MathSciNet)
MR1658686

Zentralblatt MATH identifier
0916.60044

Subjects
Primary: 60G42: Martingales with discrete parameter
Secondary: 60H05: Stochastic integrals

Rights

#### Citation

Montgomery-Smith, Stephen. Concrete Representation of Martingales. Electron. J. Probab. 3 (1998), paper no. 15, 15 pp. doi:10.1214/EJP.v3-37. https://projecteuclid.org/euclid.ejp/1454101775

#### References

• D.J. Aldous, Unconditional bases and martingales in Lp(F), Math. Proc. Cambridge Philos. Soc. 85, (1979), 117-123.
• D.L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9, (1981), 997-1011.
• C. Dellacherie and P.-A. Meyer, Probabilities and Potential, North-Holland Mathematics Studies 29, North-Holland, Amsterdam-New York-Oxford, 1978.
• S. Kwapien and W.A. Woyczynski, Tangent sequences of random variables, in Almost Everywhere Convergence, G.A. Edgar and L. Sucheston, Eds., Academic Press, 1989, pp. 237-265.
• T.R. McConnell, A Skorohod-like representation in infinite dimensions, Probability in Banach spaces, V (Medford, Mass., 1984), 359-368, Lecture Notes in Math., 1153, Springer, Berlin-New York, 1985.
• T.R. McConnell, Decoupling and stochastic integration in UMD Banach spaces, Probab. Math. Statist. 10, (1989), 283–295.