A law of the iterated logarithm for the number of blocks in regenerative compositions generated by gamma-like subordinators

TL;DR

This paper establishes a law of the iterated logarithm for the number of blocks in certain random compositions generated by gamma-like subordinators, extending understanding of their asymptotic behavior.

Contribution

It introduces a law of the iterated logarithm for block counts in compositions derived from gamma-like subordinators, with an auxiliary result on Brownian motion convolutions.

Findings

Law of the iterated logarithm for block counts as n→∞

Asymptotic behavior of compositions from gamma-like subordinators

New result on Brownian motion and regularly varying functions

Abstract

The points of the closed range of a drift-free subordinator with no killing are used for separating into blocks the elements of a sample of size $n$ from the standard exponential distribution. This gives rise to a random composition of $n$. Assuming that the subordinator has the L\'{e}vy measure, which behaves near zero like the gamma subordinator, we prove a law of the iterated logarithm for the number of blocks in the composition as $n$ tends to infinity. Along the way we prove a law of the iterated logarithm for the Lebesgue convolution of a standard Brownian motion and a deterministic regularly varying function. This result may be of independent interest.