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Two equivalence methods for sequences of random variables will be discussed in some detail. The first of the two methods is based on a study of measures on an infinite product space and projections of these measures on certain distinguished sub-sigma-rings of sets in the space. This method will be applicable to sequences of dependent random variables as well as to independent random variables. The limit laws falling under the scope of this method are those for which the conclusion is a property which is to be true almost everywhere for the sequence of random variables. The second method will treat only the case of independent random variables; however, it will apply to the central limit theorem as well as to the common examples of the laws indicated above. This method will depend on the vertical distance between pairs of distribution functions, while the first method, in the case of independent random variables, depends on the slopes of the distribution functions.