Summary
LetX=(X 1, ...,X d ) be an ℝd Lévy process on ℝ+ or ergodic exchangeable process on [0, 1], and letV=(V 1, ...,V d ) be a predictable process on the same interval. Under suitable moment conditions, it is shown that, if the Lebesgue integrals\(\int {\prod\limits_{j \in J} {V_j } } \) are a.s. non-random for allJ⊂{1, ...,d} with #J≦d-1 or #J≦d, respectively, then the product moment EΠ∫V j dX j is the same as ifX andV were independent. An analogous statement holds in discrete time. The results imply some invariance properties of exchangeable sequences and processes under suitable predictable transformations
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References
Aldous, D.J.: Exchangeability and related topics. In: Hennequin, P.L. (ed.) École d'été de probabilités de Saint-Flour XIII-1983. (Lect. Notes Math., vol. 1117, pp. 1–198) Berlin Heidelberg New York: Springer 1985
Chow, Y.S., Teicher, H.: Probability theory. New York Berlin Heidelberg: Springer 1978
Dellacherie, C., Meyer, P.A.: Probabilités et potentiel, Chapters I-VIII. Paris: Hermann 1975/80
Doob, J.L.: Note on probability. Ann. Math.37, 363–367 (1936)
Franken, P., Lisek, B.: On Wald's identity for dependent variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.60, 134–150 (1982)
Ideda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam Tokyo: North-Holland & Kodansha 1981
Jacod, J.: Calcul stochastique et problèmes de martingales. Lect. Notes Math., vol. 714. Berlin Heidelberg New York: Springer 1979
Kallenberg, O.: Canonical representation and convergence criteria for processes with interchangeable increments. Z. Wahrscheinlichkeitstheor. Verw. Geb.27, 23–36 (1973)
Kallenberg, O.: Path properties of processes with independent and interchangeable increments. Z. Wahrscheinlichkeitstheor. Verw. Geb.28, 257–271 (1974a)
Kallenberg, O.: Series of random processes without discontinuities of the second kind. Ann. Probab.2, 729–737 (1974b)
Kallenberg, O.: On the existence and path properties of stochastic integrals. Ann. Probab.3, 262–280 (1975)
Kallenberg, O.: Characterizations and embedding properties in exchangeability. Z. Wahrscheinlichkeitstheor. Verw. Geb.60, 249–281 (1982)
Kallenberg, O.: Some surprises in finite gambling and its continuous time analogue. In: Lanke, J., Lindgren, G. (eds.) Contributions to probability and statistics in honour of Gunnar Blom, pp. 205–214. Lund: Studentlitteratur 1985
Kallenberg, O.: Spreading and predictable sampling in exchangeable sequences and processes. Ann. Probab.16, 508–534 (1988)
Karatzas, I., Shreve, S.: Brownian motion and stochastic calculus. New York Berlin Heidelberg: Springer 1988
Klass, M.J.: A best possible improvement of Wald's equation. Ann. Probab.16, 840–853 (1988)
Wald, A.: Sequential tests of statistical hypotheses. Ann. Math. Statist.16, 117–186 (1945)
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Research supported by NSF grant DMS-8703804
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Kallenberg, O. General wald-type identities for exchangeable sequences and processes. Probab. Th. Rel. Fields 83, 447–487 (1989). https://doi.org/10.1007/BF01845699
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DOI: https://doi.org/10.1007/BF01845699