On the uniform convergence of random series in Skorohod space and representations of c\`{a}dl\`{a}g infinitely divisible processes

Let $X_n$ be independent random elements in the Skorohod space $D([0,1];E)$ of c\`{a}dl\`{a}g functions taking values in a separable Banach space $E$. Let $S_n=\sum_{j=1}^nX_j$. We show that if $S_n$ converges in finite dimensional distributions to a c\`{a}dl\`{a}g process, then $S_n+y_n$ converges a.s. pathwise uniformly over $[0,1]$, for some $y_n\in D([0,1];E)$. This result extends the It\^{o}-Nisio theorem to the space $D([0,1];E)$, which is surprisingly lacking in the literature even for $E=R$. The main difficulties of dealing with $D([0,1];E)$ in this context are its nonseparability under the uniform norm and the discontinuity of addition under Skorohod's $J_1$-topology. We use this result to prove the uniform convergence of various series representations of c\`{a}dl\`{a}g infinitely divisible processes. As a consequence, we obtain explicit representations of the jump process, and of related path functionals, in a general non-Markovian setting. Finally, we illustrate our results on an example of stable processes. To this aim we obtain new criteria for such processes to have c\`{a}dl\`{a}g modifications, which may also be of independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AOP783 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The role of education in sustainable development

Texts referring to the sustainable development of organisations strongly emphasise the perspective of digitisation of the organisation’s functioning. The text indicates the current models of educational organisations, based on the typology by F. Laloux and indicates their adequacy to individual stages of the industrial revolution. Then, analysing these business models, one of the models in the context of sustainable development of the organisation, indicates the specificity of demand. The author also indicates one of the possible ways to obtain the desired model of the functioning of an educational organisation – presenting an appropriate case study. The theoretical background of the text was created on the basis of the analysis of the F. Laloux model and its reference to the realities of educational organisations. The types of organisation characteristic of the second, third and fourth industrial revolutions are indicated. While the types characteristic of the second and third industrial revolutions are widespread, the types characteristic of the fourth industrial revolution can take two forms. The author proposes one of them as more adequate to the challenges of education for sustainable development. The methodology of the presented choice of the type of organisation is based on a case study prepared on the basis of interviews with stakeholders of the educational institution

The minimum wage in the national economy: reasons and changes in Poland

Motivation: The minimum wage protect employees from excessively low wages. It helps ensure fair participation in economic development for the general public. The minimum wage in the economy is an essential element in a country’s economic policy. Decisions concerning the implementation and changes in the minimum wage are part of the socio-economic policy of the state. Research on minimum wage relate to its overall impact on the level and structure of wages, employment, productivity at work, the size of the gray economy, the level of poverty and public finances. Factors affecting changes in the minimum wage are formal and non-formal. The question therefore arises which factors are decisive in the Polish economy. Aim: The aim of this paper is to present the essence and causes of changes in the minimum wage in Poland in 2003–2020 with particular regard to formal and non-formal aspects. Results: On the basis of research carried out in Poland, a link between minimum wage and inflation was observed. There is also a fundamental relationship between the minimum wage and average wages in the economy. It has been shown that the minimum wage is not without significance for the impact on public finance sector in Poland

The negative organisational consequences of average leadership skills : case studies based on Ken Blanchard's paradigm

Background. Organizational leadership is constantly present in the reflection of management science. It seems that the timeliness of the organizational leadership topic comes not only from the importance of managing teams of workers, but also from the changes that are taking place in: attitudes employees, the expectations of the organization and in the organizational environment. It causes the need for continuous reflection, because solutions adequate a couple of years ago, may not be conformed to contemporary situation. Organizational leadership is an interesting theme of reflection, because is an area of knowledge between management science and psychology. It is possible to venture the statement, that the reflection from one of these areas is enriched by the second one. Research aims. An average level of managerial skills can be considered as “good enough”. However, does this mean that it is the same as “correct”? The article indicates that nowadays “average” is “too little”. Methodology. On the basis of a few selected case studies of Polish organisations, the author presents the disadvantages of an average level of managerial skills. The analysis of the case studies and the particular types of organisational disorders will be based on the Situational Leadership Model developed by Ken Blanchard. The author analyses two situations in organisations: the leader using mainly styles S2 and S3 and the leader using mainly styles S1 and S4. Key findings. The author shows the consequences of average level of managerial skills for: a single employee, the team, the effects for another organisational processes (recruitment, employee turnover). The article also describes the psychological mechanism (social game or the archetype) that is often the background of that kind of behaviour (S2-S3; S1-S4) in the organisation

Group theoretic dimension of stationary symmetric \alpha-stable random fields

The growth rate of the partial maximum of a stationary stable process was first studied in the works of Samorodnitsky (2004a,b), where it was established, based on the seminal works of Rosi\'nski (1995,2000), that the growth rate is connected to the ergodic theoretic properties of the flow that generates the process. The results were generalized to the case of stable random fields indexed by Z^d in Roy and Samorodnitsky (2008), where properties of the group of nonsingular transformations generating the stable process were studied as an attempt to understand the growth rate of the partial maximum process. This work generalizes this connection between stable random fields and group theory to the continuous parameter case, that is, to the fields indexed by R^d.Comment: To appear in Journal of Theoretical Probability. Affiliation of the authors are update

Asymptotic independence of multiple Wiener-It\^{o} integrals and the resulting limit laws

We characterize the asymptotic independence between blocks consisting of multiple Wiener-It\^{o} integrals. As a consequence of this characterization, we derive the celebrated fourth moment theorem of Nualart and Peccati, its multidimensional extension and other related results on the multivariate convergence of multiple Wiener-It\^{o} integrals, that involve Gaussian and non Gaussian limits. We give applications to the study of the asymptotic behavior of functions of short and long-range dependent stationary Gaussian time series and establish the asymptotic independence for discrete non-Gaussian chaoses.Comment: Published in at http://dx.doi.org/10.1214/12-AOP826 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org