Abstract We study higher analogues of the classical independence number on $\omega $ . For $\kappa regular uncountable, we denote by $i(\kappa )$ minimal size a maximal -independent family. establish ZFC relations between and standard some cardinal characteristics, e.g., $\mathfrak {r}(\kappa )\leq \mathfrak {i}(\kappa {d}(\kappa measurable, assuming that $2^{\kappa }=\kappa ^{+}$ construct fam...

This paper studies the interrelations between independence or conditional independence and causal relations, defined in terms of functional dependence, that hold among variables of interest within the settable system framework of White and Chalak. We provide formal conditions ensuring the validity of Reichenbach’s principle of common cause and introduce a new conditional counterpart, the condit...

Mutual stochastic independences among-algebras and mutual algebraic inde-pendences among elements of semimodular lattices are observed to have a very similar behaviour. We suggest abstract independence structures called I-relations describing it. Presented examination of I-relations resembles a theory of abstract connectedness: a dual characterization of I-relations by families of connected set...

 Many Limit Theorems, Convergence Theorems and Characterization Theorems in Probability and Statistics, in particular those related to normal distribution , are based on the assumption of independence of two or more random variables. However, the full power of independence is not used in the proofs of these Theorems, since it is the distribution of summation of the random variables whic...