Theoretical justification of computing the 3-base periodicity using nucleotide distribution variance

In a previous paper (Yin and Yau, 2005), a novel method was proposed to measure the power spectrum of a DNA sequence at frequency N/3 in order to distinguish protein-coding and non-coding regions in DNA sequences. This was accomplished by computing the distribution of the four nucleotides in the three reading frames (codon positions) and identifying variance as an indicator of 3-base periodicity. That work included an empirical justification for the claim that there exists a linear, 3:2 correlation between the variance and the power spectrum. In this note, we provide a theoretical justification for that observation in the form of a mathematical proof of this correlation. This work thus provides a more rigorous justification for the use of the variance instead of the more computationally expensive power spectrum, allowing users of this technique to apply it with absolute confidence that no compromise in accuracy is incurred.