Cumulative record times in a Poisson process

Abstract

We obtain a strong law of large numbers and a functional central limit theorem, as t → ∞, for the number of records up to time t and the Lebesgue measure (length) of the subset of the time interval [0, t] during which the Poisson process is in a record lifetime.

Keywords:

• almost sure representations
• functional central limit theorem
• Poisson process
• record value
• renewal process
• strong law of large numbers

AMS Subject Classification:

• Primary 60G70
• secondary 60F15
• 60F17
• 60K05