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On Positive Random Objects

Author

Listed:
  • Johan Jonasson

    (Chalmers University of technology)

Abstract

The idea of defining the expectation of a random variable as its integral with respect to a probability measure is extended to certain lattice-valued random objects and basic results of integration theory are generalized. Conditional expectation is defined and its properties are developed. Lattice valued martingales are also studied and convergence of sub- and supermartingales and the Optional Sampling Theorem are proved. A martingale proof of the Strong Law of Large Numbers is given. An extension of the lattice is also studied. Studies of some applications, such as on random compact convex sets in R n and on random positive upper semicontinuous functions, are carried out, where the generalized integral is compared with the classical definition. The results are also extended to the case where the probability measure is replaced by a σ-finite measure.

Suggested Citation

  • Johan Jonasson, 1998. "On Positive Random Objects," Journal of Theoretical Probability, Springer, vol. 11(1), pages 81-125, January.
    • Handle: RePEc:spr:jotpro:v:11:y:1998:i:1:d:10.1023_a:1021694808465
    DOI: 10.1023/A:1021694808465
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    References listed on IDEAS

    as
    1. Hiai, Fumio & Umegaki, Hisaharu, 1977. "Integrals, conditional expectations, and martingales of multivalued functions," Journal of Multivariate Analysis, Elsevier, vol. 7(1), pages 149-182, March.
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