Mapping Theorem (other)
Mapping theorem may refer to * Continuous mapping theorem In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps converg ..., a statement regarding the stability of convergence under mappings * Mapping theorem (point process), a statement regarding the stability of Poisson point processes under mappings {{disambiguation ...
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Continuous Mapping Theorem
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps convergent sequences into convergent sequences: if ''xn'' → ''x'' then ''g''(''xn'') → ''g''(''x''). The ''continuous mapping theorem'' states that this will also be true if we replace the deterministic sequence with a sequence of random variables , and replace the standard notion of convergence of real numbers “→” with one of the types of convergence of random variables. This theorem was first proved by Henry Mann and Abraham Wald in 1943, and it is therefore sometimes called the Mann–Wald theorem. Meanwhile, Denis Sargan refers to it as the general transformation theorem. Statement Let , ''X'' be random elements defined on a metric space ''S''. Suppose a function (where ''S′'' is another metric space) has the set of discontinu ...
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