Hidden Words Statistics for Large Patterns
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We study here the so called subsequence pattern matching also known as hidden pattern matching in which one searches for a given pattern w of length m as a subsequence in a random text of length n. The quantity of interest is the number of occurrences of w as a subsequence (i.e., occurring in not necessarily consecutive text locations). This problem finds many applications from intrusion detection, to trace reconstruction, to deletion channel, and to DNA-based storage systems. In all of these applications, the pattern w is of variable length. To the best of our knowledge this problem was only tackled for a fixed length m=O(1) [Flajolet, Szpankowski and Vallée, 2006]. In our main result we prove that for m=o(n^1/3) the number of subsequence occurrences is normally distributed. In addition, we show that under some constraints on the structure of w the asymptotic normality can be extended to m=o(√(n)). For a special pattern w consisting of the same symbol, we indicate that for m=o(n) the distribution of number of subsequences is either asymptotically normal or asymptotically log normal. We conjecture that this dichotomy is true for all patterns. We use Hoeffding's projection method for U-statistics to prove our findings.
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