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.
We introduce the notions of asymptotic quasi-inverse functions and as-ymptotic inverse functions as weaker versions of (quasi-)inverse functions, and study their main properties. Asymptotic quasi-inverse functions exist in the class of so-called pseudo-regularly varying (PRV) functions, i.e. functions preserving the as-ymptotic equivalence of functions and sequences. On the other hand, asymptotic inverse functions exist in the class of so-called POV functions, i.e., functions with positive order of variation. In this paper, we obtain some new results about PRV and POV functions. Some further properties of asymptotic (quasi-)inverse functions as well as some applications will be discussed in Part II of this paper to appear in no. 71 of this journal.
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