Genomic aberrations, such as somatic copy number alterations, are frequently observed in tumor tissue. Recurrent aberrations, occurring in the same region across multiple subjects, are of interest because they may highlight genes associated with tumor development or progression. A number of tools have been proposed to assess the statistical significance of recurrent DNA copy number aberrations, but their statistical properties have not been carefully studied. Cyclic shift testing, a permutation procedure using independent random shifts of genomic marker observations on the genome, has been proposed to identify recurrent aberrations, and is potentially useful for a wider variety of purposes, including identifying regions with methylation aberrations or overrepresented in disease association studies. For data following a countable-state Markov model, we prove the asymptotic validity of cyclic shift $p$-values under a fixed sample size regime as the number of observed markers tends to infinity. We illustrate cyclic shift testing for a variety of data types, producing biologically relevant findings for three publicly available datasets.