Memoryless Properties on Time Scales

ABSTRACT

The classical memoryless property is well-known to induce the geometric distribution for discrete random variables and the exponential distribution for continuous random variables. When the range of the random variable is just an arbitrary closed subset of the real line, significant difficulties arise as to the interpretation of the memoryless property itself. Multiple proposals have been made, and we explore their consequences. In particular, we consider whether a given definition of “memoryless” will induce a specific distribution.