Probability limits can be used if theta follows say a gamma or beta distribution, but I am not sure of using probability limit approach for some charts like EWMA or CUSUM. I would like to know if control limit of the form E(theta) +/- L*SD(theta) is always applicable when theta is not normally distributed.
An individual control chart is not robust to non-normally distributed data. So you would need to transform theta or at the very least remove outliers.
See: Gest. Prod., São Carlos, v. 23, n. 1, p. 146–164, 2016 http://dx.doi.org/10.1590/0104–530X1445-14, Method for determining the control limits of nonparametric charts for monitoring location and scale.
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the answer will indeed depend on your perspective : if your perspective is statistically-oriented, then, of course, non normal data should not be treated as if they were normal and there is a continuous flow of litterature on this topic ... but it is more relevant to the theory of statistical tests as opposed to the practice of control charts
shewhart should not be confused with neyman and if your perspective is that of a decision-maker using control charts to understand and improve his/her process, then the answer may be quite different
the following posts are very interesting (and provocative) if your perspective is that of a decision-maker :
http://www.qualitydigest.com/inside/quality-insider-article/myths-about-process-behavior-charts.html
http://www.qualitydigest.com/inside/twitter-ed/are-you-sure-we-don-t-need-normally-distributed-data.html
http://www.qualitydigest.com/inside/quality-insider-column/don-t-we-need-remove-outliers.html
http://www.qualitydigest.com/inside/quality-insider-column/process-behavior-charts-non-normal-data-part-1.html
http://www.qualitydigest.com/inside/quality-insider-column/process-behavior-charts-non-normal-data-part-2.html
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Its not about the normality. Its all about the known mean and variance of your statistic (say theta).
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