Abstract.A general method for obtaining the strong law of large numbers for sequences of random variables is considered. Some applications for dependent summands are given.Key words. strong law of large numbers, Hájek-Rényi maximal inequality, ρ-mixing, logarithmically weighted sums
PII. S0040585X97978385Introduction. The strong law of large numbers (SLLN) asserts that a sequence of cumulative sums of random variables becomes "nonrandom" by normalizing it by an appropriate sequence of nonrandom numbers and approaching the limit. Many results of this type were obtained for both independent and dependent summands forming cumulative sums.There are two basic approaches to proving the strong law of large numbers. The first is to prove the desired result for a subsequence and then reduce the problem for the whole sequence to that for the subsequence. In so doing, a maximal inequality for cumulative sums is usually needed for the second step. Note that maximal inequalities make up a well-developed branch of probability theory and many inequalities are known for different classes of random variables.The second approach is to use directly a maximal inequality for normed sums. Inequalities of this kind are said to be of Hájek-Rényi type, referring to the paper by Hájek and Rényi [5] devoted to independent summands. Inequalities of this type are not easy to obtain and the first approach prevails. However, after a Hájek-Rényi inequality is obtained, the proof of the strong laws of large numbers becomes an obvious problem. Some Hájek-Rényi inequalities were announced in [8].In this paper, our goals are to show that a Hájek-Rényi type inequality is, in fact, a consequence of an appropriate maximal inequality for cumulative sums and to show that the latter automatically implies the strong law of large numbers. Most important, we made no restriction on the dependence structure of random variables. To reach these goals we prove two basic theorems. Several examples of applications are given in separate sections. We do not consider orthogonal and stationary dependence structures because results in these cases are well known; however, these are two areas of possible application of our general approach. Also, we do not discuss in detail the case of independent summands; however, this case appears in several parts of the paper as an example of the origin of the theory.
There are several environmental issues in urban areas that are caused by the unintentional consequences of past activities. One of these issues is the wide application of asbestos cement in roofing materials in the 2 nd half of the 1900s. In this study, our goal was to identify different roof types and to determine those with asbestos components using high-ground (1 m) and spectral (126 bands) resolution airborne hyperspectral imagery (AISA Eagle II) and several classification approaches. In addition, we aimed to identify those wavelengths that play a significant role in distinguishing the different roof types. In the image analysis, the SAM, MLC and SVM classification methods were used to evaluate the different types of roofs. These methods resulted in accurate maps of the roof types, and asbestos cement roofs were identified with over 85% accuracy.
A general random graph evolution mechanism is defined. The evolution is a combination of the preferential attachment model and the interaction of N vertices (N ≥ 3). A vertex in the graph is characterized by its degree and its weight. The weight of a given vertex is the number of the interactions of the vertex. The asymptotic behaviour of the graph is studied. Scale-free properties both for the degrees and the weights are proved. It turns out that any exponent in (2, ∞) can be achieved. The proofs are based on discrete time martingale theory.
A random graph evolution mechanism is defined. The evolution studied is a combination of the preferential attachment model and the interaction of four vertices. The asymptotic behaviour of the graph is described. It is proved that the graph exhibits a power law degree distribution; in other words, it is scale-free. It turns out that any exponent in(2,∞)can be achieved. The proofs are based on martingale methods.
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