If $T_n$ is consistent for $T$ and $U_n$ consistent for $U$, is $T_n + U_n$ consistent for $T+U$?
Secondly, is $T_n/U_n$ consistent for $T/U$?
I am looking for a reference for these facts, if they are true.
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3$\begingroup$ These are a direct consequence of Slutsky's theorem. See for example the discussion of it here. $\endgroup$– Glen_bCommented Nov 27, 2016 at 1:41
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1$\begingroup$ @Glen_b the convergence in distribution there confuses me. Don't we require convergence in probability for consistency? $\endgroup$– FlashCommented Nov 27, 2016 at 3:33
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1$\begingroup$ en.wikipedia.org/wiki/… $\endgroup$– Glen_bCommented Nov 27, 2016 at 10:28
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$\begingroup$ Are you sure? I cannot find anything related to your statement in Chung's book. $\endgroup$– JulianCommented Dec 9, 2021 at 18:01
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1$\begingroup$ I always think of this in terms of the continuous mapping theorem. $\endgroup$– Geoffrey JohnsonCommented Dec 9, 2021 at 18:05
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1 Answer
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Certainly the sum of consistent estimates are consistent. Regarding the ratio there is a need to specify that $U_n$ does not converge 0.
These results can be found in most advanced probability books including K. L. Chung's, or at Wikipedia.
answered Nov 26, 2016 at 17:07