Conditional Stable Laws and Rare-Event Limits for Absorbing Markov Chains

Abstract

We establish conditional limit theorems, pointwise in the initial state, for absorbing Markov chains on a compact metric space $M$. We assume $L^1(M,ρ)$-continuous transition densities, irreducibility and aperiodicity. For the observable $f_β(x)=d_M(x,x_0)^{-β}$, with suitable $x_0$ satisfying $ρ(B_r(x_0))\sim C_d(x_0)r^d$, we prove that the point-process of normalised observations converges to a Poisson random measure. This yields totally right-skewed $α$-stable laws for $α=d/β\in(0,2)$ and, at the boundary value $α=2$, a Gaussian limit with the non-standard normalisation $\sqrt{n\log n}$. We also establish a conditional central limit theorem for $L^2$ observables, exponential deviation bounds for bounded observables and a conditional Poisson law for visits to shrinking targets.