Asymptotic Expansions in Probability Theory
UNIVERSITY OF KIEV KIEV Ukraine
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The natural solution to many problems in probability theory and mathematical statistics is provided by an examination of certain limit distributions. As is the case in the classical problem of the summation of a large number of random variables which are in some sense of equal weight, the study of the exact distribution functions of the sums not only leads, as a rule, to intractable formulas but in many important cases is impossible, since the exact distributions of the separate summands are often unknown. On the other hand, limit distributions are almost independent of the idiosyncrasies of the distributions of the summands and have a quite manageable form. The same phenomenon can be observed in mathematical statistics in the study of statistical criteria for a large number of observations. As a rule, the exact criteria are complicated but their limiting form is simple and convenient for application. An excellent example of this situation is the treatment of statistical mechanics by A. I. Khinchin, in which limit theorems play an outstanding role. Examples of this kind are literally innumerable.
- Statistics and Probability