A ratio ergodic theorem for Brownian additive functionals with infinite mean
Let \(B_t\), \(t\geq 0\), be a real-valued Brownian motion and \(L(x,t)\), \(x\in R^1\), \(t\geq 0\), its local time. Consider two additive functionals \[ A^\pm(t) =\int^\infty_0 L(\pm x,t) dm^\pm(x), \quad t\geq 0, \] where \(m^\pm(x)\), \(x\in (-\infty, \infty)\), are two non-negative non-decreasing right-continuous functions with \(m^\pm (0)=0\) such that \(dm^\pm\) both have support in \([0,\infty)\). An integral test is proposed to investigate the asymptotic behaviour of the ratio \(k(t)=A^+ (t)/A^-(t)\) in the case when \(m^\pm (\infty)= \infty\), i.e. when the standard ratio ergodic theorem does not apply. It is proved that \(k^*= \limsup_{t\to\infty} k(t)= \infty\) (or 0) according to the divergence (convergence) of the integral.
scientific article; zbMATH DE number 1059872