On a problem connected with quadratic regression

Laha, R. G.; Lukács, Eugene

doi:10.1093/biomet/47.3-4.335

Cited by 96 publications

(42 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An important direction of such investigations was relaxing the assumptions of independence of U and V . The same characterization holds true when instead of assuming independence, constancy of the first and second conditional moment of U given by V is assumed, see [5] or [14] for more general so called Laha-Lukacs regressions. One can also consider other powers of U (see eg.…”

Section: Introductionmentioning

confidence: 82%

See 1 more Smart Citation

On the Lukacs property for free random variables

Szpojankowski

2015

Studia Math.

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 82%

“…From this we obtain that after applying the Lemma 2.2 every maximal sequence of neighbouring IY has as a left neighbour X −1 Y. In particular if in the left hand side of (14) there is sequence of l neighbouring IY then it must be of the form…”

Section: The Lukacs Property In Free Probabilitymentioning

confidence: 94%

On the Lukacs property for free random variables

Szpojankowski

2015

Studia Math.

View full text Add to dashboard Cite

show abstract

“…Remark 7 (Laha-Lukacs relations). In [LL60], Laha and Lukacs proved that the Meixner distributions are characterized by a certain property involving linear conditional expectations and quadratic conditional variances. In [BB06], Bożejko and Bryc proved that an identical characterization holds, in the free setting, for the free Meixner distributions.…”

Section: Free Meixner Distributions and Conditional Freenessmentioning

confidence: 99%

Free evolution on algebras with two states

Anshelevich

2010

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

The key result in the paper concerns two transformations, Φ : (ρ, ψ) → ϕ and B t : ψ → ϕ, where ρ, ψ, ϕ are states on the algebra of non-commutative polynomials, or equivalently joint distributions of d-tuples of non-commuting operators. These transformations are related to free probability: if ⊞ is the free convolution operation, and {ρ t } is a free convolution semigroup, we show that. The maps {B t } were introduced by Belinschi and Nica as a semigroup of transformations such that B 1 is the bijection between infinitely divisible distributions in Boolean and free probability theories. They showed that for γ t the free heat semigroup and Φ the Boolean version of the Kolmogorov representation for infinitely divisible measures,The more general map Φ [ρ, ψ] comes, not from free probability, but from the theory of two-state algebras, also called the conditionally free probability theory, introduced by Bożejko, Leinert, and Speicher. Orthogonality of the c-free versions of the Appell polynomials, investigated in [Ans08b], is closely related to the map Φ. On the other hand, more general free Meixner families behave well under all the transformations above, and provide clues to their general behavior. Besides the evolution equation, other results include the positivity of the map Φ [ρ, ψ] and descriptions of its fixed points and range.

show abstract

“…In this paper we will be particularly interested in characterizations of probability measures by regression assumptions. An example of such characterization are well known Laha-Lukacs regressions [12] which characterize so called Meixner distributions by assumption that for independent X and Y first conditional moment of X given by X + Y is a linear function of X + Y and second conditional moment of the same type is a quadratic function of X + Y . In [4,6] authors studied free analogues of Laha-Lukacs regressions, which characterize free Miexner distribution defined in [2].…”

Section: Introductionmentioning

confidence: 99%