New characterization of two-state normal distribution

In this article we give a purely noncommutative criterion for the characterization of two-state normal distribution. We prove that families of two-state normal distribution can be described by relations which is similar to the conditional expectation in free probability, but has no classical analogue. We also show a generalization of Bozejko, Leinert and Speicher's formula (relating moments and noncommutative cumulants).Comment: 19 pages, 2 figures, accepted for publication by Infinite Dimensional Analysis, Quantum Probability and Related Topic

Characterizations of some free random variables by properties of conditional moments of third degree polynomials

We investigate Laha-Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Levy processes. The proof relies on a combinatorial identity. At the end of this paper we show that this result can be extended to a q-Gausian variable.Comment: Journal of Theoretical Probability, 201

Nowe oblicze wartości księgowej spółek giełdowych w dobie światowych kryzysów gospodarczych

The main issue addressed in the article relates to the impact of complex economic crises on the book value of joint-stock companies. Turmoil on the stock market has a negative impact on the book value of the entities studied. This is more important because investors buying or selling shares of individual listed companies have a difficult task, since in making decisions around their investments they are exposed to enormous risk and, at the same time, uncertainty. In this case, it is worth examining which stock market indicators are most helpful in assessing the financial health of companies. The purpose of this article is to determine the impact of the selected indicators on the growth of book value of the companies listed on the Warsaw Stock Exchange during crisis circumstances. The research method used in the presented article is the estimation of an econometric model that confirms the high impact of selected metrics on the book value of listed companies The research on the impact of estimators on the dependent variable covers the period January 2020–May 2023.Zasadnicza problematyka poruszana w artykule odnosi się do wpływu złożonych kryzysów gospodarczych na wartość księgową spółek akcyjnych. Zawirowania na giełdzie papierów wartościowych mają negatywny wpływ na wartość księgową badanych jednostek. Jest to o tyle istotne, że inwestorzy skupujący bądź zbywający akcje poszczególnych spółek giełdowych mają utrudnione zadanie, gdyż podejmując decyzje wokół realizowanych inwestycji są narażeni na ogromne ryzyko a zarazem niepewność. Warto w tym wypadku zbadać, które wskaźniki giełdowe są najbardziej pomocne w ocenie kondycji finansowej spółek. Celem niniejszego artykułu jest określenie wpływu wybranych wskaźników na wzrost wartości księgowej spółek notowanych na Giełdzie Papierów Wartościowych w Warszawie w warunkach kryzysu. Metodą badawczą zastosowaną w prezentowanym artykule jest estymacja modelu ekonometrycznego potwierdzającego wysoki wpływ wybranych mierników na wartość księgową spółek akcyjnych Badania nad wpływem estymatorów na zmienną zależną obejmują przedział czasowy styczeń 2020–maj 2023

Sample Variance in Free Probability

Let $X_1, X_2,\dots, X_n$ denote i.i.d.~centered standard normal random variables, then the law of the sample variance $Q_n=\sum_{i=1}^n(X_i-\bar{X})^2$ is the $\chi^2$-distribution with $n-1$ degrees of freedom. It is an open problem in classical probability to characterize all distributions with this property and in particular, whether it characterizes the normal law. In this paper we present a solution of the free analog of this question and show that the only distributions, whose free sample variance is distributed according to a free $\chi^2$-distribution, are the semicircle law and more generally so-called \emph{odd} laws, by which we mean laws with vanishing higher order even cumulants. In the way of proof we derive an explicit formula for the free cumulants of $Q_n$ which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of $R$-cyclicity.Comment: Final version to appear in J of Funct Anal; 24 pages;Corollary 4.14 generalized; gap in the proof of Prop 4.13 fixe

The Boolean quadratic forms and tangent law

In \cite{EjsmontLehner:2020:tangent} we study the limit sums of free commutators and anticommutators and show that the generalized tangent function $$\frac{\tan z}{1-x\tan z}$$ describes the limit distribution. This is the generating function of the higher order tangent numbers of Carlitz and Scoville \cite[(1.6)]{CarlitzScoville:1972} which arose in connection with the enumeration of certain permutations. In the present paper we continue to study the limit of weighted sums of Boolean commutators and anticommutators and we show that the shifted generalized tangent function appears in a limit theorem. In order to do this, we shall provide an arbitrary cumulants formula of the quadratic form. We also apply this result to obtain several results in a Boolean probability theory.Comment: 28 page