Asymptotic laws for compositions derived from transformed subordinators

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A random composition of n appears when the points of a random closed set R⊂[0,1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts Kₙ of this composition and other related functionals under the assumption that R=ϕ(S), where (Sₜ,t≥0) is a subordinator and ϕ:[0,∞]→[0,1] is a diffeomorphism. We derive the asymptotics of Kₙ when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function ϕ(x)=1-e⁻ˣ, we establish a connection between the asymptotics of Kₙ and the exponential functional of the subordinator.