A predominance of certain ‘sequence systemmetries’ in DNA/RNA has led to various conjectures about the possible structural/functional role these symmetries might play in nucleic acid sequences. De Wachter employed a binomial probability model to compare the observed number of ‘direct repeats’ with those expected in a random sequence. Counting of direct repeats essentially leads to a sequence of m-dependent trials. We develop a stochastic model for studying various types of symmetries. Expressions for means and variances of the statistics employed are derived. The asymptotic distributions are obtained using the central limit theorem for m-dependent random variables. It is proposed that each sequence pattern be examined separately for its chance occurrence as opposed to what de Wachter suggests, i.e., clumping of all patterns together. It is also shown how our model can be used to detect various gene-amplification events, if any, in nucleic acid sequences. Finally, for certain types of patterns, it is indicated how the theory of recurrent events can be used to get a better handle on the analysis of direct repeats.