Continuous mapping theorem

If ${\displaystyle X_{n}}$ are random elements with values in a metric space and ${\displaystyle X_{n}{\xrightarrow {\mathcal {D}}}\,X}$, ${\displaystyle h}$ is a function on the metric space, and the probability that ${\displaystyle X}$ attains a value where ${\displaystyle h}$ is discontinuous is zero, then ${\displaystyle h(X_{n}){\xrightarrow {\mathcal {D}}}h(X)}$ (^[1] page 31, Corollary 1, ^[2] page 21, Theorem 2.7)

This includes for example the convergence of the sum of two sequences of random variables ${\displaystyle X_{n}}$ and ${\displaystyle Y_{n}}$ (the random element is the pair of the random variables, the continuous function is the mapping of the pair to the result of the operation), but only in the case where

${\displaystyle (X_{n},Y_{n})\,{\xrightarrow {\mathcal {D}}}\,(X,Y).}$

We note that this does not lead to a more general case of Slutsky's Theorem, because that would require only the assumption

${\displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X}$ and ${\displaystyle Y_{n}\,{\xrightarrow {\mathcal {D}}}\,Y,}$

which does not imply ${\displaystyle (X_{n},Y_{n})\,{\xrightarrow {\mathcal {D}}}\,(X,Y)}$, so we cannot apply the Continuous mapping theorem.

References

Notes

1. Billingsley, Patrick (1969). Convergence of Probability Measures. John Wiley & Sons. ISBN 0471072427
2. Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons. p. 2nd edition. ISBN 0471197459