Gaussian Approximation via Martingale Methods

Gordin’s seminal paper (1969) initiated a line of research in which limit theorems for stationary sequences are proved via appropriate approximations by stationary martingale difference sequences followed by an application of the corresponding limit theorem for such sequences. In this chapter, we first review different ways to get suitable martingale approximations and then derive the central limit theorem and its functional form for strictly stationary sequences under various types of projective criteria. More general normalizations than the traditional ones will be also investigated, as well as the functional moderate deviation principle. We shall also address the question of the functional form of the central limit theorem for not necessarily stationary sequences. The last part of this chapter is dedicated to the moderate deviations principle and its functional form for stationary sequences of bounded random variables satisfying projective-type conditions.

Representações dos proprietários e funcionários de fazendas sobre as mudanças e conservação da vegetação ciliar às margens do rio São Francisco, Nordeste do Brasil

In Brazil, areas with high water potential for development ofirrigate agriculture was marked by the occupation of human groups. In this context, the vegetation along the São FranciscoRiver is constantly changing. This study sought to answer the following questions: “How do famers and farmworkersdescribe the possible changes in riparian forest over time?”; “Which events are responsible for these possible changes?”,and; “How do farmers describe aspects related to the use and conservation of riparian vegetation?” A total of 17 men andeight women, owners and farmworkers from areas close to the riparian vegetation, in five municipalities in the states ofBahia and Pernambuco, participated in this research. Semi-structured and semi-structured projective-type interviewswere used to investigate their representations of the conservation of riparian vegetation and changes in the local landscape,and to record historical events that influenced them. Aiming to record the local knowledge about the diversity of the mostimportant useful species, the free list technique was used. Only one informant said that the riparian vegetation has notchanged, six considered that the changes were for the worse and four considered that the changes were for the better. Fourowners responded that changes in the vegetation began 10 years ago, six indicated more than 30 years ago, two did notknow and one has not perceive any changes. Amongst the farmworkers, five said that vegetation had changed more than20 years ago, four indicated more than 10 years ago and four indicated that it had not changed. All informants agreed thatthe riparian vegetation should be preserved either because it protects the river (64%) or because it has other uses, such asattracting rain, providing shade, medicinal uses and increasing oxygen. Some of the informants (48%) considered the landowners responsible for the conservation of riparian vegetation, some (48%) that this was a duty of everybody and asmaller portion (4%) attributed responsibility to IBAMA. They also pointed out solutions to the conservation of thisvegetation: reforestation (39%), non-clearance (17%), environmental education (13%) and good care (13%). Futurerestoration projects that might take place in this area should take into account these representations, demands andexpectations.

Projective Geometry and Collisions

A few years ago a series of papers by Frank Coffman Bell was published in this Journal suggesting how collision avoidance manoeuvres could be analysed using projective geometry. However it is probably true that most non-mathematicians have little experience of any type of geometry other than the Euclidean metric geometry which is taught in schools. Euclid's approach was to elaborate a science of the measurement of physical space and for this purpose he deduced a number of geometrical theorems starting from the intuitional ideas of point, line, distance and length. To Euclid and his followers the notion of distance was completely obvious and basic and underlay everything in geometrical science, hence the terminology that is given to this work ‘metric geometry’. Another concept of geometry was however put forward by a group of geometers of whom Pappus is one of the best known. His work shows that he was interested in theorems concerned not with distance but with such things as concurrence of lines and collinearity of points. This type of theorem may be thought of as the projective type. If for example it is assumed that any two lines in a plane meet in a point, then two parallel lines must also meet in a point which may be termed a point at infinity. In projective geometry no distinction is then made between ordinary points and points at infinity.