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.
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.