Mathematics of the USSR-Sbornik · 1976 · 162 citations · 3 references
Natural SciencesMathematicsProbability TheoryStatisticsStochastic ProcessesMetric GeometryMetric SpacesProbability DistributionsProbability SpacesRandom VariablesDistance Metrics
The principle of three stages in the study of approximation problems, particularly distribution approximation, is formulated. The paper introduces a metric on the space of random variables and presents results on stability in distribution characterization and remainder estimation for sums of independent variables. Metrics are defined on random variables in general spaces (metric, Banach, Hilbert) and used to analyze the three‑stage approximation process, with proofs of related facts. The authors prove various facts about metrics in the three‑stage approximation and provide results on stability and remainder terms for limiting distributions of sums of independent random variables. Bibliography contains 11 titles.
In this paper the concept of a metric in the space of random variables defined on a probability space is introduced. The principle of three stages in the study of approximation problems is formulated, in particular problems of approximating distributions. Various facts connected with the use of metrics in these three stages are presented and proved. In the second part of the paper a series of results is introduced which are related to stability problems in characterizing distributions and to problems of estimating the remainder terms in limiting approximations of distributions of sums of independent random variables. Both the account of properties of metrics and the application of these facts in the second part of the paper are presented under the assumption that the random variables take values in a general space (metric, Banach or Hilbert space). Bibliography: 11 titles.
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