Large deviation estimates related to arcsine laws for subordinators

TL;DR

This work addresses large deviation estimates for the Dynkin–Lamperti limit $X_{T(s)-}/s$ of subordinators, extending the classical Beta/arcsine limit to explicit rate estimates. It employs regular variation of the Laplace exponent $\Phi$ and the double Laplace transform to derive asymptotics in both long-range ($\lambda\to0+$) and short-range ($\lambda\to\infty$) regimes. The main contributions are two theorems giving asymptotic forms $\mathbb{P}(X_{T(s)-}/s\le c(s)) \sim \frac{\sin(\pi\alpha)}{\pi\alpha}\frac{\ell(s)}{\ell(c(s)s)} c(s)^{\alpha}$ (long-range) and its short-range analogue, under precise regular variation assumptions and with explicit control via slowly varying functions. The results provide explicit large-deviation rates for the Dynkin–Lamperti arcsine-type limits, with potential applications in excursion theory and fluctuation theory for subordinators.

Abstract

We establish large deviation estimates related to the Dynkin--Lamperti theorem, which is a distributional limit theorem for the position of a subordinator immediately before it crosses a fixed level. Our approach relies on the theory of regular variation and the method of the double Laplace transform.

Theorems & Definitions (12)

• Example 1.1
• Theorem 2.1: Long-range Dynkin--Lamperti theorem
• Remark 2.2
• Theorem 2.3: Short-range Dynkin--Lamperti theorem
• Theorem 3.1: Large deviation estimates related to the long-range Dynkin--Lamperti theorem
• Remark 3.2
• Remark 3.3
• Remark 3.4: Comparison with the long-range Dynkin--Lamperti theorem
• Theorem 3.5: Large deviation estimates related to the short-range Dynkin--Lamperti theorem
• Lemma 4.1: Double Laplace transform of $X_{T(t)-}$ Ber