Some weighted approximations in probability of self-normalized and Studentized partial sums processes are reviewed and also described in the context of studying the problem of change in the mean of random variables in the domain of attraction of the normal law. This survey of such results constitutes an extended abstract of the talk with the same title that was presented by Miklos Csorgő on July 18, 2005 in Mikulov, based on the joint works M. Csorgő, B. Szyszkowicz, and Q. Wang (2001), (2003) and (2004) by the three of us.
This paper investigates weighted approximations for studentized $U$-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition $E|h(X_1,X_2)|^2 < \infty$ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.