Continuous Mapping Theorem

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In probability theory, the continuous mapping theorem states that continuous functions are limit- preserving 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 {xn} with a sequence of random variables {Xn}, and replace the standard notion of convergence of real numbers “->” with one of the types of convergence of random variables.

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