Talk:Continuous mapping theorem

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)

Continuity of g
The statement of the theorems (correctly) ask for the set of discontinuity points Dg to be a measure-zero set, i.e., that g is almost-surely continuous. However, the assumption that g is continuous was used for the proofs. Notice that the proof for the convergence in distribution becomes more involved when the assumption of a.s. continuity of g is used [cf. Theorem 2.7 ].

On a related note, in the proof of convergence in probability, it must be shown that Bδ is indeed measurable.