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An operator-valued free Poincarè inequality

Ito Hyuga

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 2

241--246

The purpose of this short note is to give an operator-valued free Poincarè inequality, which provides a simple proof to (an improvement of) a lemma of Voiculescu (2000) asserting that the kernel of the free difference quotient is exactly the coefficients.

On limit theorems in JW-algebras

Karimov A., Mukhamedov F.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 1

153--165

In the present paper, we study bundle convergence in JW- algebra and prove certain ergodic theorems with respect to such convergence. Moreover, conditional expectations of reversible JW-algebras are considered. Using such expectations, the convergence of supermartingales is established.

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 distributions of conditional expectations

Paszkiewicz A.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 2

287--291

Let F and G be distribution functions on R. Then there exist a random variable X and a σ-field U satisfying P(X < a) = F(a), P(E(X|U) < a) = G(a) .[formula]. The consideration is kept on a rather elementary level.

Conditioning and weak convergence

Jajte R., Paszkiewicz A.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 2

453--461

Several connections between the weak convergence of random variables, convergence of their distributions and conditioning have been described.