In this paper, we investigate noncommutative symmetric and asymmetric maximal inequalities associated with martingale transforms and fractional integrals. Our proofs depend on some recent advances on algebraic atomic decomposition and the noncommutative Gundy decomposition. We also prove several fractional maximal inequalities.

This paper is devoted to the study of some asymptotic properties of aM-estimator in a framework of detection of abrupt changes inrandom field's distribution. This class of problems includes e.g.recovery of sets. It involves various techniques, including M-estimation method, concentrationinequalities, maximal inequalities for dependent random variables and ϕ-mixing. Penalization of the criterion function when the size of thetrue model is unknown is performed. All the results apply under mild, discussedassumptions. Simple examples are provided.

Conditional stochastic ordering is concerned with the stochastic ordering of a pair of probability measures conditional on certain subsets or sub-σ -algebras. Some basic results of conditional stochastic ordering were proved by Whitt. We extend some of Whitt's results and prove a basic relation between stochastic ordering conditional on subsets and stochastic ordering conditional on σ -algebras. In the second part of the paper we consider the ordering of conditional expectations. There are several different formulations of this problem motivated by different types of applications.