Mathematical Statistics by Johann Pfanzagl

Author:Johann Pfanzagl
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

Proof

We have

Now for and for and , and, with , for . It follows that for .

A Remark on Continuous Convergence

Let be a metric space with metric , and let , , be a sequence of functions.

Definition 5.3.10

The convergence of the sequence to is continuous at if implies .

Continuous convergence was used in certain parts of mathematics (see e.g. Hahn 1932). It was introduced in asymptotic statistical theory by Schmetterer (1966). To motivate this step (see p. 301), he does not say more thanit seems better to introduce the idea of continuous convergence. When the limit of a sequence of functions is continuous, the idea of continuous convergence is even more general [?] than the idea of uniform convergence.

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