On the approximation of a random variable by a conditional expectation of another random variable

• Autorzy: Kaniowski K.
• Języki publikacji: EN

Abstrakty

EN

Let X and Y be R-valued random variables on a non-atomic probability space (Ω, F, P). We give conditions under which Y can be approximated by a conditional expectation of X. In particular, we prove the following theorem: Let X be an R-valued random variable such that EX⁺ = EX⁻ = ∞. Then for each random variable Y and arbitrary ϵ > 0 there exist B ∈ F and a sub-σ-field S of F such that P (B) ≤ ϵ and E (X|S) = Y a.s. on B^c. We also review some facts on the conditional expectation of unintegrable random variables.

Słowa kluczowe

EN

conditional expectation

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2005
• Tom: Vol. 25, Fasc. 2
• Strony: 393--403
• Opis fizyczny: Bibliogr. 5 poz.

Twórcy

autor

Kaniowski K.

• Faculty of Mathematics, University of Łódź, Banacha 22, 90-238 Łódź, Poland

Bibliografia

• [1] B. Fristedt and L. Gray, A Modern Approach to Probability Theory, Birkhäuser, Boston-Basel-Berlin 1997.
• [2] O. Kallenberg, Foundations of Modern Probability, Springer, 1997.
• [3] K. Kaniowski, On the sequences whose conditional expectations can approximate any random variable, Probab. Math. Statist. 22 (2002), pp. 115-126.
• [4] K. Kaniowski, On the stochastic convergence of conditional expectations of some random sequences, Ann. Univ. Mariae Curie-Skłodowska, Sect. A, 56 (2002), pp. 47-52.
• [5] R. E. Strauch, Conditional expectations of random variables that do not necessarily have finite expectations, Ann. Math. Statist. 36 (1965), pp. 1556-1559.