  In this note, we establish the convergence in distribution of the maxima of
i.i.d. random variables to the Gumbel distribution with the associated
normalizing sequences for several examples that are related to the normal
distribution. Motivated by tests for jumps in high-frequency data, our main
interest is in the half-normal distribution and the sum or difference of two
independent half-normally distributed random variables. Since the half-normal
distribution is neither stable nor symmetric, these examples are non-obvious
generalizations. It is shown that the sum and difference of two independent
half-normally distributed random variables and other examples yield
distributions with tail behaviours that relate to the normal case. It turns out
that the Gumbel convergence for all such distributions can be proved following
similar steps. We illustrate the results in Monte Carlo simulations.
