Talk:Continuous mapping theorem
|WikiProject Statistics||(Rated B-class, Mid-importance)|
|WikiProject Mathematics||(Rated B-class, Mid-importance)|
This theorem has a generalization to differences. For convergence in probability, it is as follows: if and are sequences of random variables such that converges to zero in probability, , and converges to in distribution, then converges to zero in probability. This is Corollary 2 in the paper by Mann & Wald.
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.