Pseudo-martingales

• Autorzy: Jajte R. , Paszkiewicz A.
• Języki publikacji: EN

Abstrakty

EN

For a probability space (Ω, ℱ, P)and a filtration (U_n) in Ω,we consider the sequences (X_n) of random variables satisfying the condition E (X_n+1 − X_n | U_n) = 0, n = 1, 2, ... In general, the proces (X_n) is not required to be (U_n) adapted and it is called a pseudo-martingale. We indicate simple and natural conditions implying a good asymptotic behaviour of pseudo-martingales. For example: let (X_n, U_n) be a uniformly integrable pseudo-martingale with U_n ↗ ℱ. Then X_n → X weakly in L₁ where [formula]. Some approximation results for σ-fields are obtained with implications to pseudo-martingales. A number of examples is collected.

Słowa kluczowe

EN

random variables; pseudo-martingale; law of the iterated logarithm

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 1999
• Tom: Vol. 19, Fasc. 1
• Strony: 181--201
• Opis fizyczny: Bibliogr. 6 poz.

Twórcy

autor

Jajte R.

• Department of Probability and Statistics, Łódź University, Banacha 22, 90-238 Łódź, Poland

autor

Paszkiewicz A.

• Department of Probability and Statistics, Łódź University, Banacha 22, 90-238 Łódź, Poland

Bibliografia

• [1] T. Ando, Contractive projections in Lp-spaces, Pacific J. Math. 17 (1966), pp. 391-405.
• [2] R. B. Ash, Real Analysis and Probability, Academic Press, New York-San Francisco-London 1972.
• [3] R. L, Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley, Massachusetts, 1994.
• [4] E. Hensz-Chądzyńska, R. Jajte and A. Paszkiewicz, Almost sure convergence of projections in Lp-spaces, Ann. Univ. Mariae Curie-Skłodowska 61 (1997), pp. 67-84.
• [5] E. H. Lacey, The Isometric Theory of Classical Banach Spaces, Die Grundlehren der Math. Wissenschaften 208, Springer-Verlag, Berlin-Heidelberg-New York 1974.
• [6] M. M. Rao, Stochastic Processes: General Theory, Kluwer Acad. Publishers, Dordrecht-Boston-London 1995.