Abstract. Long-range dependent random fields with spectral densities which are unbounded at some frequencies are investigated. We demonstrate new examples of covariance functions which do not exhibit regular varying asymptotic behaviour at infinity. However, variances of averaged functionals of these fields are regularly varying. Limit theorems for weighted functionals of cyclical long-range dependent fields are obtained. The order of normalizing constants and relations between the weight functions and singularities in non-degenerative asymptotics are discussed.MSC 2010. 60G60, 60F17

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SummaryFreeman has considered the following two-stage procedure for finding a confidence interval for the treatment difference theta, using data from an AB/BA crossover trial. In the first stage, a preliminary test of the null hypothesis that the differential carryover is zero, is carried out. If this hypothesis is accepted then the confidence interval for theta is constructed assuming that the differential carryover is zero.If, on the other hand, this hypothesis is rejected then this confidence interval is constructed using only data from the first period. Freeman has shown that this confidence interval has minimum coverage probability far below nominal. He therefore concludes that this confidence interval should not be used. In the present paper, we analyse the performance of a similar two-stage procedure for an ABAB/BABA crossover trial. This trial differs in very significant ways from an AB/BA crossover trial, including the fact that for an ABAB/BABA crossover trial there is an unbiased estimator of the differential carryover that is unaffected by between-subject variation. Despite these great differences, we arrive at the same conclusion as Freeman.Namely, that the confidence interval resulting from the two-stage procedure should not be used.