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4 Kaniowski K.

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

Kaniowski K.

Probability and Mathematical Statistics

2005

Vol. 25, Fasc. 2

393--403

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 Bc. We also review some facts on the conditional expectation of unintegrable random variables.

On the sequences whose conditional expectations can approximate any random variable

Kaniowski K.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 1

115--126

Let(Ω,F, P)be a non-atomic probability space. For a given sequence (Xn)of random variables we indicate a number of conditions whichimply that for anyrandom variable Y there exists a sequence(Un) of σ-fields satisfying E(Xn|Un)→Ya.s. In particular, we formulate a sufficient condition using the distributions of Xn’sonly.

On the approximation of a random variable by a conditioning of a given sequence

Kaniowski K.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 2

321--328

Let (Ω,F, P) be a non-atomic probability space. If (Xn) is a sequence of r.v.’s satisfying Xn → 0 a.s. (respectively, in probability) as n→∞ and EX+n→∞, EX-n→ ∞, as n → ∞, then for any r.v. Y there exists a sequence (Un) of σ-fields such that E(Xn|Un|)→Ya.s. (respectively, in probability) as n→∞.

On some connections between random partitions of the unit segment and the Poisson process

Kaniowski K.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 2

373--381

Let Dt be the diameter of a partition of the interval [0, t] by renewal moments of a standard Poisson process. Then Dt/lnt → 1 for t →∞, in probability. Other theorems on diameters are obtained. Jajte’s theorem on random partitions of the unit segment is used.