Continuous mapping theorem: Difference between revisions

Revision as of 02:49, 9 November 2015

Revision as of 02:51, 9 November 2015

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===Convergence in probability===

Fix an arbitrary ''ε'' > 0. Then for any ''δ'' > 0 consider the set ''B_δ'' defined as

: <math>

B_\delta = \big\{x\in S\ \big|\mid x\notin D_g:\ \exists y\in S:\ |x-y|<\delta,\, |g(x)-g(y)|>\varepsilon\big\}.

</math>

This is the set of continuity points ''x'' of the function ''g''(·) for which it is possible to find, within the ''δ''-neighborhood of ''x'', a point which maps outside the ''ε''-neighborhood of ''g''(''x''). By definition of continuity, this set shrinks as ''δ'' goes to zero, so that lim_{''δ''→0&nbsp;→&nbsp;0}''B_δ'' &nbsp;= &nbsp;∅.

Now suppose that |''g''(''X'') &nbsp;− &nbsp;''g''(''X_n'')| &nbsp;> &nbsp;''ε''. This implies that at least one of the following is true: either |''X''−''X_n''|&nbsp;≥&nbsp;''δ'', or ''X''&nbsp;∈&nbsp;''D_g'', or ''X''∈''B_δ''. In terms of probabilities this can be written as

: <math>

\operatorname{Pr}\big(\big|g(X_n)-g(X)\big|>\varepsilon\big) \leq

\operatorname{Pr}\big(|X_n-X|\geq\delta\big) + \operatorname{Pr}(X\in B_\delta) + \operatorname{Pr}(X\in D_g).

</math>

On the right-hand side, the first term converges to zero as ''n'' &nbsp;→ &nbsp;∞ for any fixed ''δ'', by the definition of convergence in probability of the sequence {''X_n''}. The second term converges to zero as ''δ'' &nbsp;→ &nbsp;0, since the set ''B_δ'' shrinks to an empty set. And the last term is identically equal to zero by assumption of the theorem. Therefore the conclusion is that

: <math>

\lim_{n\to\infty}\operatorname{Pr} \big(\big|g(X_n)-g(X)\big|>\varepsilon\big) = 0,

</math>

which means that ''g''(''X_n'') converges to ''g''(''X'') in probability.