Geometrically distributed Stirling words and Stirling compositions

A nonempty word w of finite length over the alphabet of positive integers is a Stirling word if for each letter i in w all entries between two consecutive occurrences of i (if these exist) are larger or equal to i. We derive an exact and also an asymptotic formula for the probability that a random geometrically distributed word of length n is a Stirling word. We also determine an asymptotic estimate for the number of compositions (called Stirling compositions) that satisfy this property. Moreover, we find generating functions and asymptotics formulas for statistics in Stirling compositions and geometrically distributed Stirling words, such as the number of distinct values and the size of the maximum part. The proofs make use of various techniques of advanced asymptotic analysis, including Mellin transforms and the saddle point method.