Asymptotic results for jump probabilities associated to the multiple scan statistic

The concept of pattern arises in many applications comprising experimental trials with two or more possible outcomes in each trial. A binary scan of type r / k is a special pattern referring to success–failure strings of fixed length k that contain at least r -successes, where r , k are positive integers with r ≤ k . The multiple scan statistic W t , k , r is defined as the enumerating random variable for the overlapping moving windows occurring until trial t which include a scan of type r / k . In the present work, we consider a sequence of independent binary trials with not necessarily equal probabilities of success and develop upper bounds for the probability of the event that the multiple scan statistic will perform a jump from ℓ to ℓ + 1 (where ℓ is a nonnegative integer) in a finite time horizon.