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If a sequence of measurable mappings defined from a measure spaces to another measure space converges in different modes (see Wikipedia and John Cook's site), I wonder if there are some concepts capturing their rates of convergence for different modes of convergence?

You may restrict the discussion to probability theory. One example is in statistics. Suppose $\theta_n$ is an estimator for $\theta^*$ based on a sample with sample size $n$. I was wondering how the rate of convergence of $\theta_n$ is defined?

I also appreciate if there can be some references offered.

Thanks and regards!

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Modes of convergence – Jean-Victor Côté Feb 16 '12 at 15:10
Thanks! How about rate of convergence? – Tim Feb 16 '12 at 15:19

There is a paper that I do not have access to that you might find relevant: Davis, R.A. (1982). The rate of convergence in distribution of the maxima, Statistica Neerlandica 36, 31-35.

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