How does choice of norm/metric effect Stochastic convergence?

I am currently working my through a Stochastic Modeling course (textbook: Introduction to Stochastic Search and Optimization by Spall). In my last lecture, we reviewed the various convergences for Stochastic Processes, e.g. in Probability, almost surely, etc.

The text claims that the convergence is invariant under the choice of norm, but I am curious as to the computational effects of different norm choices. For instance, given an epsilon, how does the choice of norm effect the probability of being epsilon close for some number of terms, etc? I haven’t been able to find many resources on these sorts of question. Any guidance is appreciated, thanks!

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Nevin Manimala

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