New developments of source coding by means of generalized exponentials with scale invariance.

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Stochastic independence is one of the important properties to characterize the standard probability theory. In order to generalize the independence including the standard one as a special case, especially for long memory dependence, this project aims the generalization of law of large numbers which is the basis of information source coding by applying scaled exponential function (q-exponential function). As the result, we clarify the relation to the q-product, naturally derived from q-exponential function, and obtain some candidates of generalization of law of large numbers. However, such an attempt requires not only generalization of law of large numbers but also other related theorems on probability theory, which needs more time and careful discussions. Thus, this project does not yet obtain a generalization of information source coding theorem withinthree years and is still under study.