Strong Limit For Two Kinds Of Blockwise Dependent Random Fields

The random variables or stochastic processes which come from practical problems are usually not independent. There has been a great amount of work on the properties of dependent random variables, and some significant results have been obtained through deep research by many scholars. In recent years, some scholars have weakened the relationship of dependent to blockwise dependence. The concepts of blockwise independent and blockwise M dependent were proposed in 1987. The definition of the latter one allows that the random variables in different blocks can be arbitrarily dependent, thus weakening the whole condition of dependence. At present, there is little result for blockwise dependent random variables of limit theory, but we find that it can be more concise and more convenient to describe the limit theory of dependent random variable. If it is applied to limit theory, we will obtain profound theoretical and practical significance, and the limit theory can have wide applications.Based on precedent research on limit theory of dependent and blockwise dependent random variables sequences, we focus on studying the question on the strong laws of large number and the order of growth of two kinds of blockwise dependent random variables---blockwiseρ~* -mixing and blockwise M -orthogonal random fields. The main contents of this paper are outlined as follows:1. Using the maximum moment inequality ofρ~* -mixing random variables and the method of summability we obtain the strong laws of large numbers for blockwiseρ~* -mixing random fields under suitable conditions, thereby extend some results in the literature form independent random fields.2. We use the similar method studying the strong laws of large numbers for blockwise M -orthogonal random fields. In addition, we also discuss the sufficient conditions of limit theory for blockwise M -orthogonal random fields. Particularly, it also shows that there are the best possible sufficient conditions for multi-indexed independent real-valued random variables.3. By the Menshov-Rademacher strong limit theorem and a theorem from Martikainen (1986) which generalizes Abel's lemma of the blockwise M -orthogonal random fields, we study the order of growth of the estimate of partial sums for this sequence.4. We use the method of truncation for the double indexes blockwise M -orthogonal random variables to discuss the Marcinkiewicz-Zygmund type law of large numbers under proper conditions, and also to expand one-dimension interval into two-dimensional arbitrary rectangular area, making results more applicable.