# Is there a statistical application that requires strong consistency?

I was wondering if someone knows or if there exists an application in statistics in which strong consistency of an estimator is required instead of weak consistency. That is, strong consistency is essential for the application and the application would not work with weak consistency.

• No, there is no such application. Oct 29, 2013 at 20:58
• Sometimes I wonder if even weak consistency - outside its intuitive appeal - is in reality very important. If I have an estimator that behaves very sensibly at every finite sample size below $n=10^{1000}$ and in reality my biggest sample size will only ever be a miniscule fraction of that, I might have an inconsistent estimator that's nevertheless perfectly fine. It seems to me that the actual value in consistency at all is that it's usually (in practical cases rather than pathological ones) associated with estimators that still behave 'nicely' as sample sizes move past what we might ever see. Nov 1, 2013 at 23:23

## 1 Answer

If you need a reference for the answer in my comment above, here is one, from Andrew Gelman's blog:

Which reminds me of Lucien Le Cam’s reply when I asked him once whether he could think of any examples where the distinction between the strong law of large numbers (convergence with probability 1) and the weak law (convergence in probability) made any difference. Le Cam replied, No, he did not know of any examples. Le Cam was the theoretical statistician’s theoretical statistician, so there’s your answer.

One could maybe add that the real importance of this different modes of convergence lies in the mathematics, that they permit the use of different mathematical techniques, in the development of the theory, only. And that might be important enough, but for the development of theory, not in the concrete practical applications.

• +1 The difference would be important, perhaps, for Buzz Lightyear, who could get to infinity--or at least knew he was headed there.
– whuber
Nov 1, 2013 at 21:13