On ℓ-overlapping Runs of Ones of Length k in Sequences of Independent Binary Random Variables

Abstract

Consider a finite sequence of independent binary (zero-one) random variables ordered on a line or on a circle. The number of the ℓ-overlapping runs of ones of a fixed length k is studied for both types of the concerned ordering. Recurrences for the exact probability mass functions for these numbers are obtained via simple probabilistic arguments. Exact closed formulae, for the mean and variance of the studied numbers are obtained via their representations through properly defined indicators. Two application case studies, concerning record sequences and reliability of consecutive systems, clarify further the theoretical results.

Keywords:

• Binary trials
• Runs
• Overlapping counting
• Exact distributions
• Exact moments
• Bounds and approximations
• Records
• System reliability

Mathematics Subject Classification:

• 60E05
• 62E15
• 60K99
• 65B05
• 62N05