In the previous chapter, we examined abstractly, four modes of convergence: convergence a.c., convergence in probability, L p convergence and Convergence in distribution; in addition, we had also explored the manner in which they are related to each other, and we had given conditions under which we may obtain convergence a.c., convergence in probability, or L p convergence for sequences of random variables. These last results, however, had been obtained on the assertion that the sequence(s) to which they were applied consisted of independent random variables. Moreover, the implicit framework of that discussion was one of scalar random variables. While many of the proofs easily generalize or, more appropriately, are applicable without modification, to sequences of random vectors, for some this is not the case. Specifically, in the proof of Proposition 11 and its related corollaries (in Chapter 3), we made explicit use of the natural order of the number system, and this does not lend itself, easily, to generalization in cases where we deal with entities more complicated than scalar random variables.
KeywordsManifold Covariance Assure Sine
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