Renewal Theory for Several Patterns

The results presented in this paper were motivated by a study of certain patterns, known as restriction sites, in DNA sequences. DNA sequences can be thought of as finite words over a four-letter alphabet {A, T, G, C}, and restriction sites are (sequence-specific) locations of positions in double-stranded DNA where it is cut by a protein known as a restriction enzyme. See Watson (1977). For example, the restriction enzyme Hpa I1 cuts at occurrences of CCGG while Hha I cuts at GCGC. Such enzymes, under proper conditions, cut the DNA at all non-overlapping occurrences of the restriction sites. The enzymes are applied alone (a single digest), in batches of two (a double digest), and so on. The goal was, then, to derive the distribution of fragment lengths when a specific set of enzymes was used to digest DNA. We assumed the DNA was an independent, identically distributed sequence of letters where the letter prob abilities were p., p, p a , p c . A direct application of Theorem 3 and the corollary to DNA sequences appears in Waterman (1983). To simplify the current discussion we use the canonical sequence of heads and tails (H and T), and generalize the renewal theory of Feller (1968). The earliest work we can locate which considers such problems is Ledlie (1967) who treats the recurrent event E(k, g) which occurs at the end of a group of k