Tight bounds and approximations for scan statistic probabilities for discrete data (Q1176973)

The sum of the last \(m\) variables in a sequence of independent and identically distributed ones is assumed and the upper and lower bounds of the distribution function of the waiting time (for the event that the mentioned sum is large as some \(k\)) are derived. The proof uses a generalization of Janson's result on continuous-time processes and of Hoover's generalization of Bonferroni-type inequalities. Results are applied to two problems. The first one treats the probability of ``almost'' perfect matching in two amino acid sequences, i.e. matching except of some a priori given numbers of positions. The second one considers the probability of the largest possible charge in scanning interval when the sequence of letters (acidic, neutral, basic) and their charges in the amino acid chain is classified. The results are improvements of results of many other authors, as Erdős, Révész, Karlin among others.