In this paper, the ordering properties of convex and increasing convex orders of the dependent random variables are studied. Some closure properties of the convex and increasing convex orders under independent random variables are extended to the dependent random variables under the Archimedean copula. Two applications are provided to illustrate our results.

We introduce and investigate several variants of a bidirectional string assemblingsystem, which is a computational model that generates strings from copies of assemblyunits. The underlying mechanism is based on two-sided piecewise assembly of adouble-stranded sequence of symbols, where the upper and lower strand have to match. Thegenerative capacities and the relative power of the variants are our main interest. Inparticular, we prove that bidirectional string assembling system generate languages notrepresented as any finite concatenation of one-sided string assembling systems. The latterbuild an infinite, strict and tight concatenation hierarchy. Moreover, it is shown thateven the strongest system in question can only generate NL languages, while there areunary regular languages that cannot be derived. Furthermore, a finite strict hierarchywith respect to the different variants considered is shown and closure properties of thelanguages generated are presented.

We introduce and investigate string assembling systems which form a computational model that generates strings from copies out of a finite set of assembly units. The underlying mechanism is based on piecewise assembly of a double-stranded sequence of symbols, where the upper and lower strand have to match. The generation is additionally controlled by the requirement that the first symbol of a unit has to be the same as the last symbol of the strand generated so far, as well as by the distinction of assembly units that may appear at the beginning, during, and at the end of the assembling process. We start to explore the generative capacity of string assembling systems. In particular, we prove that any such system can be simulated by some nondeterministic one-way two-head finite automaton, while the stateless version of the two-head finite automaton marks to some extent a lower bound for the generative capacity. Moreover, we obtain several incomparability and undecidability results as well as (non-)closure properties, and present questions for further investigations.

We consider shifted equality sets of the form EG(a,g1,g2) = {ω | g1(ω) = ag2(ω)}, where g1 and g2 are nonerasing morphisms and a is a letter. We are interested in the family consisting of the languages h(EG(J)), where h is a coding and (EG(J)) is a shifted equality set. We prove several closure properties for this family. Moreover, we show that every recursively enumerable language L ⊆ A* is a projection of a shifted equality set, that is, L = πA(EG(a,g1,g2)) for some (nonerasing) morphisms g1 and g2 and a letter a, where πA deletes the letters not in A. Then we deduce that recursively enumerable star languages coincide with the projections of equality sets.

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.