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This paper studies the problem of testing the null assumption of no-change inthe mean of chronologically ordered independent observations on a randomvariable $X$ {\it versus} the at most one change in the mean alternativehypothesis. The approach taken is via a Darling-Erd\H{o}s type self-normalizedmaximal deviation between sample means before and sample means after possibletimes of a change in the expected values of the observations of a randomsample. Asymptotically, the thus formulated maximal deviations are shown tohave a standard Gumbel distribution under the null assumption of no change inthe mean. A first such result is proved under the condition that $EX^2\log\log(|X|+1)<\infty$, while in the case of a second one, $X$ is assumed to be in aspecific class of the domain of attraction of the normal law, possibly withinfinite variance.

Change in the mean in the domain of attraction of the normal law via Darling-Erd\H{o}s theorems