# Talk:Continuous mapping theorem

WikiProject Statistics (Rated B-class, Mid-importance)

This article is within the scope of the WikiProject Statistics, a collaborative effort to improve the coverage of statistics on Wikipedia. If you would like to participate, please visit the project page or join the discussion.

B  This article has been rated as B-Class on the quality scale.
Mid  This article has been rated as Mid-importance on the importance scale.
WikiProject Mathematics (Rated B-class, Mid-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 B Class
 Mid Importance
Field: Probability and statistics

## Generalization

This theorem has a generalization to differences. For convergence in probability, it is as follows: if $(X_n)$ and $(Y_n)$ are sequences of random variables such that $X_n - Y_n$ converges to zero in probability, $\lim_{n \to \infty} Pr[X_n \in D_g] = 0$, and $Y_n$ converges to $Y$ in distribution, then $f(X_n) - f(Y_n)$ converges to zero in probability. This is Corollary 2 in the paper by Mann & Wald.

--Kaba3 (talk) 22:39, 5 November 2014 (UTC)

## Quantifier for set of discontinuity points

The phase in the theorem that says "has the set of discontinuity points Dg such that Pr[X ∈ Dg] = 0 " would be clearer if the quantifier for the set Dg was made plainer. I think the condition Pr[X ∈ Dg] = 0" is intended to apply to each discontinuity point of the function. If so, the intended meaning is that each discontinuity point of the function satisfies the condition. However another interpretation of "has the set of discontinuity points" is that there exists a set of discontinuity points that satisfy the condition. By that interpretation, Dg need not contain all the discontinuity points.

Tashiro (talk) 17:19, 31 May 2011 (UTC)