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

We prove a log average almost-sure invariance principle (log asip) for renewal processes with positive i.i.d. gaps in the domain of attraction of an $\alpha$-stable law with $0<\alpha<1$. 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 {FisherTalet2011}. 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 {AaronsonDenkerFisher1992} and of {ChungErdos1951}, motivated by fractal geometry.