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

We establish conditional limit theorems, pointwise in the initial state, for absorbing Markov chains on a compact metric space $M$. We assume $L^1(M,\rho )$-continuous transition densities, irreducibility and aperiodicity. For the observable $f_{\beta }(x)=d_M(x,x_0{)}^{-\beta }$, with suitable $x_0$ satisfying $\rho (B_r(x_0))\sim C_d(x_0)r^d$, we prove that the point-process of normalised observations converges to a Poisson random measure. This yields totally right-skewed $\alpha$-stable laws for $\alpha =d\mathrm{/}\beta \in (0,2)$ and, at the boundary value $\alpha =2$, 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.