Asymptotic self-similarity and order-two ergodic theorems for renewal flows

Abstract

We prove a log average almost-sure invariance principle (abbreviated log asip) for renewal processes with positive i.i.d. gaps in the domain of attraction of an α-stable law with 0 < α < 1. Dynamically, this means that renewal and Mittag-Leffler paths are forward asymptotic in the scaling flow, up to a time average. This strengthens the almost-sure invariance principle in log density we proved in [FT11]. The scaling flow is a Bernoulli flow on a probability space. We study a second flow, the increment flow, transverse to the scaling flow, which preserves an infinite invariant measure constructed using singular cocycles. A cocycle version of the Hopf Ratio Ergodic Theorem leads to an order-two ergodic theorem for the Mittag-Leffler increment flow. Via the log asip, this result then passes to a second increment flow, associated to the renewal process. As corollaries, we have new proofs of theorems of [ADF92] and of [CE51], motivated by fractal geometry.